- Email: [email protected]
Manpower allocation in U.S. postal facilities: A heuristic approach
Compnr.& Opr Ra.. Vol. 4. pp. 257-269.
PergamonPress. 1977
Printed in Greai Britrm
MANPOWER ALLOCATION IN U.S. POSTAL FACILITIES: A HEURISTIC APPROACH M. J. S~OWAL~R,* Quantitative
Analyses ~pa~ment
College of Business Administration Madison, Wisconsin, U.S.A.
University
of Wisconsin, Madison,
and L. J. KRMEwsKIIand L. P. Faculty of Management
RITZMANS
Sciences College of Administrative Sciences, The Ohio State University, Ohio 43210. U.S.A. (Received
May
Columbus,
1976)
Scope and purpose-The purpose of this article is to explore the importance of scheduling the manpower resource in a service organization. The ability to schedule manpower to meet variations in demand can have a significant impact on the level of service that an organization can provide. This article focuses upon how to generate efficient employee work schedules for the work force in a large mail sorting facility of the U.S. Postal Service. These facilities are highly labor inten&e and service to the public depends largely upon how quickly the work force can process the mail. These mail sorting facilities are multi-stage processing systems, therefore, the exact time an employee is scheduled to work and the processing stage at which he/she is assigned largely determines how rapidly mail moves through the system. An efficient rule of thumb procedure is outlined for assigning each employee in the work force. General applicability of this procedure is relatively limited due to the specialized nature of the constraints in a mail sorting facility. Abstract-The work force tour assignment problem in a U.S. Postal Service Sectional Center Facility consists of specifying the weekly tour start time, work center assignment, and mail class responsibility for each postal employee. Any efficient work force assignment must be made subject to a number of constraints including tour schedule requirements, work center capacities, mail arrival volumes, and mail flow patterns through the processing system. A heuristic aigo~thm of a building, or cons~ction, nature is described which evaluates these many constraints in developing efficient work force tour assignment solutions. Once the problem setting and the heuristic algorithm are described, the use of the procedure is demonstrated by analyzing a hypothetical problem. INTRODUCTION
The United States Postal Service (USPS) has been besieged with problems for years, primarily in the areas of customer service and deficit budgets. These problems have taken on added significance since the reorganization on 1 July, 1971. In striving to meet the higher expectations of the public[l2], postal management can act on two different levels. The first level includes more strategic alternatives such as modernization programs, mechanization, and the separation of mail processing into two subsystems fbuik and general). The second level is the tactical one of finding better ways to allocate limited resources so as to best meet service standards. This study deals with this second level of managerial discretion. More specifically, its goal is to develop a satisfactory meth~ology for making work force tour ussign~enfs in a sec~ionul center facility (SCF). *Michael J. Showalter is Assistant Professor of Quantitative analysis, College of Business Administration, University of Wisconsin-Madison. He received his Ph.D. in Production Management from The Ohio State University. He is author of papers appearing in Monngemenl Science and Business and Economic Review, as well as a number of regional and national conferences. He is a member of TIMS and AIDS. tLeRoy J. Krajewski is Professor of Management Sciences, College of Administrative Science at the Ohio State University. He received his Ph.D. in Production Management from the University of Wisconsin-Madison. He is the author of papers appearing in Management Science, Bell Journai of Economics and Management, DeciSion Sciences, and the Proceedings of the Fourth Annual Simulation Symposium, as well as in the proceedings of a number of regional and national conferences. He is a member of TIMS and AIDS. Starry P. Ritzman is a Professor on the Faculty of Management Sciences, College of Adminstrative Science at Tbe Ohio State University. He received a D.B.A. in Production Management from Michigan State University. His papers have appeared in ~anagemenr Science, &r&ion Sciences, and P~ceedings of the Fourth Annual Simulation Symposium, as well as the proceedings of several national conferences. He is a member of TIMS, ORSA, and AIDS. 257
258
M. J.
SHOWALTERet al.
THE SCF SYSTEM
There are approximately 600 sectional center facilities located throughout the United States. Each SCF, employing between 1000 and 2000 persons, has responsibility for mail delivery and service to a particular geographic region of the country. Nonetheless, all SCFs have a set of common characteristics which are important considerations in the tour assignment problem. Mail classifications
and arrival patterns
Mail processed at an SCF is classified into several categories, based upon its origin (incoming or originating), priority and destination (local or outgoing). For example, mail can be classified into 8 different categories upon its arrival: originating or incoming airmail, originating or incoming first-class mail, originating or incoming second-class mail, or originating or incoming third-class mail. Exiting mail can similarly be classified 8 ways. Furthermore, mail enters an SCF with sizable peaks and valleys in the arrival pattern, depending on the hour of day and day of the week. In addition, these time dependent arrival patterns differ by mail classification. Mail processing
and dispatch schedules
A typical SCF has 20 identifiable work centers for processing the different mail categories. Each of the work centers represents a different kind of processing operation. Mail arriving at an SCF for processing may pass through several of these work centers before it is ready to be dispatched. The routing of mail through the work centers is a function of the mail’s priority designation, physical characteristics, and destination. Figure 1 illustrates the routing of first-class mail destined for outgoing dispatch in a typical SCF. Similar subsystems of work stations can be defined for each of the other mail classifications. Each of the nodes in Fig. 1 represents a different work center. The number within each circle is an identifying code. The decimal value associated with the entering branches represents the proportion of mail arriving at a work center which is first-class mail. The decimal value associated with the remaining branches represents the proportion of mail leaving the source node which is transferred to the next work center. Figure 1 points out clearly that an SCF is a multistage processing system. For the subsystem of 8 work centers in Fig. 1, the first stage work centers are 010,020 and 180. Stage 2 work centers are 035,060, and 080. The last stage consists of 040 and 070. It should also be noted that some work centers process more than one mail class; i.e. they appear in more than one mail subsystem. Dispatch schedules
The dispatch, or shipment, of mail from an SCF occurs according to a daily schedule. There are dispatch schedules for both local and outgoing dispatches. Over the short run these dispatch times are fairly inflexible; this is particularly true for outgoing dispatches transported by independent contractors (truck, air taxi and commercial airlines). The dispatch schedule is an 1
0 u
Fig. 1. Outgoing first-class mail subsystem.
Manpower allocation in U.S. postal facilities
259
important determinant of the quality and reliability of service. No matter how quickly mail is processed, it cannot be shipped until the next regularly scheduled dispatch. In a similar fashion, mail not completely processed prior to a first dispatch time must await the second scheduled dispatch. Other parameters
Over the intermediate time horizon of one year, the operating capacity of an SCF is relatively fixed. The processing capacity C, of work center w is limited by the number of individual work stations available. With the exception of work center 080, manpower can be added to a work center during any time span until its processing capacity limit is reached.* Another parameter which is relatively fixed in the short run is the productivity that can be expected from an employee at each center. This productivity rate remains the same regardless of the mail class; the major determinant of productivity is the type of task performed. THE TOUR ASSIGNMENT
PROBLEM
The work force tour assignment problem within an SCF can be defined as the assignment of each employee in a total work force pool (or “complement”) to a specific work week (by hour and by day) to process a specific mail class at a specific work center. Solution space and constraints
Tour assignments are constrained by the national postal labor contract and characteristics of the SCF processing system as described in the previous section. Each career regular (full-time) employee is allowed 40 hr of work per week. His work week consists of 5 consecutive 8-hr working days. The daily shift is 9 hr in duration with 1 hr allowed for meal time. The daily shift start times for all 5 consecutive work days must be identical. Postal management has complete freedom within these constraints to specify an employee’s tour assignment. Thus, an employee may begin his 5-day tour at any time of the day on any day of the week. By simply specifying a tour start time, the entire work week is defined. For example, a tour start time of Tuesday at 12 a.m. means that an employee will work from 12a.m. to 9a.m. on Tuesday through Saturday of each week. The 40 working hours so specified by a tour start time are referred to in this paper as its tour window. For the purposes of this study, the set of tour start times is restricted to 12 2-hr periods during each day. Coupled with the constraints cited earlier, this reduces the number of alternative solutions for each employee to 6720 (84 tour start times x 20 work centers x 4 mail classes). However, the problem has been specified as one of “work force” manpower tour assignments. This view of the problem reveals the combinatorial nature of the solution space. A tour assignment must be specified simultaneously for a thousand or more employees. The tour assignment of one employee has an effect on the most desirable tour assignments for all other employees. This interdependence between employee tour assignments is traced to the interdependence of the work centers in the multi-stage processing system. Objectives
The paramount objective in SCF tour assignments is to maximize service or the throughput rate of mail, given the available complement size. Service can be measured as the total volume of mail making the closest dispatch time after its arrival to the system. In addition to this objective, there is also a second objective of minimizing the number of Sunday and night manhours called for by the workforce tour assignment. More specifically, a 25% pay premium must be paid for any 8-hr shift which requires an employee to work during any portion of the time interval from 12 p.m. Saturday to 12 p.m. Sunday. Furthermore, a 10% premium is paid for all hours worked between 6 p.m. and 6 a.m. every day of the week. A third objective of management may be to limit the number of allowed daily tour start times, due to the administrative costs and confusion associated with having too many tour start times per day. *Work center 080 is mechanized with several employees manning each machine. It is highly desirable to assign a full crew to a*machine.if it is to be operated during a tour. CAOR
Vol. 4 No. 4-C
260
M.J.SHOWALTER
et al.
ALTBRNATESOLUTIONPROCEDURES
Several procedures have been reported in the literature for shift scheduling in service organizations, ranging from heuristic algorithms to optimi~tion techniques[l,2,~, IO]. Unfortunately, these procedures are designed for single-stage operations rather than the multistage SCF environment. Other reasons why these approaches are inadequate for an SCF are (1) the existence of dispatch times and (2) the existence of several objectives. The most prominent solution methodologies to consider for such a multi-stage problem are (1) optimiza~on techniques such as integer or linear programming, (2) simulation, and (3) heuristic pro~amming. Optimization techniques It is theoretically possible to represent the SCF work force tour assignment problem with a gigantic integer programming model. The objective function would be to minimize system inventory levels over the planning horizon. However, even just a one-day planning horizon results in a matrix size of 4000 by 8000. Extending the model to accommodate the necessary seven-day horizon makes such an optimization approach computationally unattractive and unwieldy. Another shortcoming is the necessity of a single objective function. Simulution One possible non-optimizing technique which is useful for evaluating complex multi-sage systems is simulation. In fact, this approach has already been developed for the SCF tour assignment problem[3]. This simulation application is based on a man-machine interaction in which the quality of solution obtained is a function of the parameter values defined by the decision maker. It allows the evaluation of tour assignments generated through trial and error. Its major limitation is its reliance on the genius of the decision maker to supply good solutions. Heuristic programming Heuristic programming is a non-optimizing methodology which offers the advantage of generating good solutions to problems. The heuristic programming approach is particularly well suited for application to illstructured problems such as tour assignment [7,8]. Rather than using trial-~d~rror procedures for generating solutions, heuristic pro~mming searches a limited set of alternatives through application of certain decision rules to develop a good solution to the problem. For the above reasons, and the inherent flexibility of heuristic programming, this methodology, coupled with simulation for evaluation purposes, has been developed for solving the SCF manpower tour assignment problem. It is described in the next section. THE HEURISTIC ALGORITHM
This algorithm can best be characterized as a “building” (or construction) heuristic procedure which makes tour assignments sequentially for more and more employees until the full complement has been assigned. The objective is to obtain a good solution without explicitly considering the existing manpower tour assignment used by the SCF. The procedure focuses upon mail arrival volumes and their timing to make tour assignments to the first-stage work centers. These assignments, coupled with routing patterns and productivities, are then used to determine the arrival volumes to the second-stage work centers, and so forth. The exact procedure is best described as consisting of the series of steps’in Fig. 2. Each of the steps is elaborated in the following sections. Specify user and system inputs-step 1 Parameter specifications can be classified as either con~ollable or uncontrollable by postal management. Certain data inputs reflect the characteristics of the processing system while other parameters may be specified by management according to their policy decisions. System parameter inputs include: (1) maximum capacity of each work center, (2) productivity rates at each work center, (3) mail arrival patterns by mail category, (4) dispatch schedules by mail category, and (5) routing patterns (see Fig. 1) for each mail category. ControlIabie p~ameter inputs subject to management prerogative include: (1) total work force size (complements for the SCF, (2) maximum allowable Sunday manhours, (3) maximum allowable night manhours,
Manpower
261
allocation in U.S. postal facilities
STEP 111
i STEP X2
SIMULATE CAPACITY TOUR ASSIGNMENT
1 INITIALIZE PROCESSING STAGE (S=l) 1
I INCRWENT STAGE (s-s+l)
-
STEP f/7
I
MARE FINAL ADJUSTMENTS
INCREMENT MAIL CLASS (m=m+l)
I
6
OUT
Fig. 2. Building heuristic program.
and (4) allowed tour start times. The last input allows management to keep adminstrative costs within limit by restricting the variety of tour start times.
Simulate capacitated
tour assignment-step 2 The second step of the algorithm is a necessary precursor to the third step which allocates the total work force complement to individual work centers. The second step begins by generating a full (or capacitated) manpower allocation by which manpower is assigned to each work center w up to its capacity limit C, for each 2-hr time period during the entire week ignoring all other constraints. Given this manpower capacitated solution, a deterministic simulation of mail flows is executed using a simulation program called DEMPSI-CAP. The output of this simulation program is the “ideal” quantity of each class of mail which makes each dispatch during the week. More importantly for the next step, it determines the ideal values for L$j, where: Lt.,,=
number of manhours of labor productivity used by the capacitated solution to process class m mail at work center w during bi-hourly time period i on day j.
Allocate complement
to work center crews-step
3
The third step is to divide the total work force complement into “work center crews”. A work center crew is defined here as the number of employees assigned to process a particular class of mail sometime during the week at a specific work center. This apportionment is begun
262
M.J.
SHOWALTER d al.
by first converting the work force complement into work center groups G,:
W6W
(1)
where: Q = work force complement size, W’ = set of work centers to which a work center group, G,, has not yet been assigned, W” = set of work centers to which a work center group, G,, has been assigned, W = W’UW”; set of all work centers, M = set of all mail classes, C, = manpower capacity of work center w. The apportionment is completed by further allocating the work center group among the various classes of mail. Thus, each work center crew, &“, is specified as follows:
where: 84’ = set of classes of mail processed at work center w for which a work center crew allocation has not yet been made, M” = set of classes of mail processed at work center w for which a work center crew allocation has been made, M = M’ U M”; set of all classes of mail. This proportional allocation process effectively reduces the manpower tour assignment problem to one of determining the tour start times for each member in each work center crew. Rank feasible fours-step 4 Steps 4 and 6 are closely interrelated. They are performed iteratively to select tours for each employee in each work center crew 6,“. With these 3 steps, the heuristic first makes a tour assignment to each first-stage work center which must process the airmail (m = 1). If this fails to assign f?,” man-tours to each work center w which processes mail m, the 3 steps are repeated until the full work center crews are assigned.? When this is completed, it is possible (1) to simulate the mail inputs into the ~eco~~-~~ffgework centers which must process the airmail (m = 1) and then (2) to apply steps 3-5 in the same fashion to make tour assignments to these second-stage work centers. This process continues until all stages are completed for the airmail. The entire procedure is then replicated for each of the Iower priority classes of mail. The only purpose of step 4 is to rank the alternate tours for each work center processing mail m in the stage being considered. The rank is essentially based on the amount of unprocessed mail, summed over all time periods in the tour window being ranked. More specifically, the rank of the tour beginning in period i* on day j at a first-stage work center is based on the value of Vci*,j where:$ tAs discussed in step 7, it may not be possible to allocate &“’ fully to all work centers. In such cases, step 7 makes the necessary final adjus~nts. $The asterisk superscript on the i indicates that the time period i is in the set of allowed tour start times.
Manpower allocationin U.S. postal facilities
263
mcFM WQWlrn i*cZ, j=1,2,...,7
(3)
and: V$*,j = total volume of unprocessed class m mail at work center w over the tour window of
the tour beginning in period i* on day j, A& = volume of class m mail arriving at the SCF in period q on day n, D”,,.q,m= number of employees assigned to process mail class m at work center w during period q and day n (due to tour assignments made at step 5 during prior iterations), I?! = weighting factor, between zero and one, which specified the importance of mail class aW,,,n m, at work center w, in period q of day n relative to its next available dispatch time,? W&VI= ELJX!,,.; output of class m mail from work center w over a feasible tour, ti+,j,which begins on interval i* on day j, Fr = proportion of class m mail arriving at the SCF which goes directly to work center w for processing, H;, = bi-hourly productivity rate per employee assigned to work center w, Z, = set of allowed tour start times, Pi = set of five consecutive days beginning with day j, Qi*= set of four consecutive two-hour time periods beginning with period i*, and WIm = set of first-stage work centers which process mail class m (a subset of w). In a similar fashion, the ranking of tour windows for work centers in stage s(s > I) is performed by replacing equation 3 with:
WEW,” i*eZf j=12 , Y-*-Y 7 s = 1,2, *. . ,ybm
(4)
where: = output of mail class m from work center I( in stage (s-t) during period 4 on day n, as ~Z?.PI determined by a simulation of mail flows given the tour assignments at stage (s-1). = set of work centers processing mail m in stage (s-Z) whose output flows (partially or ZJWrn wholly) to work center w for further processing. B,,w = proportion of class m mail flowing from work center u directly to work center W,and ybm= number of work center phases required to process class m mail for a type b dispatch. Make a tour assignment--step 5 For a given mail class m and stage s, step 5 begins by randomly choosing a work center By not restricting this choice to the tour which has one of the highest K rankings, VEi*+jn tThis allows the user to pIace greater emphasison mail which is flowing into a work center at a time at (or before) its next scheduleddispatch.
264
M.J.
SHOWALTER eral.
window with the highest ranking (allowing K > l), the heuristic algorithm gains the desirable capability of generating a variety of good work force tour assignments. The number of employees assigned to the tour so selected for a first-stage work station is equal to:
( VZ,i*,jl(2O*li,. 1)> 0 m&f WE W*m
i*df j= 1,2, . * . , 7
(9
where: @ = remaining Sunday man-tour equivalents allowed but not yet assigned, and Y = remaining night man-tour equivalents allowed but not yet assigned. After computing an Nc’*.i value for a first-stage work center, step 5 concludes by updating the values of D&,n, Y, and Q, to reflect the new tour assignment. The procedure for making tour assignments to work centers in subsequent stages (s > I) is essentially the same, except that the set of work centers evaluated is mW;” (s = 2,. . . , ybm) rather than weW;“. Simulate maii flows-step 6 For a given class m mail, all tour assignments must be made for stage (s - I) work centers
before considering stage s work centers. After stage (s - I) assignments are completed, the mail flows must be simulated to determine the output rates of all stage s work centers processing class m mail. This output information on O&, which can be obtained by a partial simulation of SCF, is then used in equations 3 and 4 to rank tour windows and make tour assignments for stage 5. The information generated by these partial simulations can also be amalgamated after the heuristic algorithm has determined the complete work force tour assignment. This information is printed out, along with the final tour assignments, at the end of the run for evaluation purposes. Make jinaf a~j~s~rnen~s-step
7
It may occur that it is impossible for step 5 to fully allocate to work center w its full work center crew SC, given the constraints on allowed tour start times, manpower capacity, allowed night manhours, and allowed Sunday manhours. When this happens the program completely eliminates the heuristically generated manpower tour assignment for that work center. The program attempts to reallocate work center crews to three consecutive, non-overlapping tours so as to maximize the IeveI of manpower allocated to that work center. To determine which non-overlapping tours will be chosen; the value V,li*,j is again calculated using either equation 3 or 4 for each feasible tour window. After choosing the maximum V,li*.i value, a manpower assignment of Nci*,i is made using equation 5 for each mail class until the capacity constraint C, is reached or work center crew is exhausted. If the capacity constraint is reached, the procedure iterates to the next feasible, non-overlapping tour start time of period (i* + 4). The maximum volume of mail (max V$+& for the set of feasible tour windows is determined and manpower assignments of N $+Jj are made for each class of mail until the capacity constraint is reached or the work center crew is exhausted. To complete
Manpower
allocation in U.S. postal facilities
265
this procedure, the same approach is utilized for the remaining non-overlapping allowed tour start time of period fi* + 8) if any work center crew remains unassigned. This procedure of non-overi~pping tour assi~ment assures that each work center crew is fully assigned and a maximum allocation is made to that work center. THE
HYPOTHETICAL
POST
OFFICE*
The post office used in this study is modelled after an SCF in the Midwest which utilizes 20 different work centers (W) in its production design ranging from opening and culling stations to secondary sorting stations. The facility processes four classes of mail: airmail, first-, second- and third-class mail. A piece of mail may pass through anywhere from two to five processing stages (s) depending upon its class and ultimate destination, In addition, the productivity rates depend upon the work center and range from 640 pieces per bi-hourly interval at the oncoming preferential opening unit to 12,030 pieces per bi-hourly interval at the culling and canceiiing station. Given a deterministic mail arrivai pattern for each mail class which varies by day of the week and hour of the day, the problem is to find feasible tour assignments for each of 800 employees in the authorized complement such that certain objectives are achieved. The building heuristic has a built-in priority structure for the objectives by relegating ah but the main objective to the constraint set. The main objective to maximize service by maximizing the volume of mail of each class which makes its first available dispatch. The di~culty in monitoring the nearest dispatch for each arriving piece of mail in a FORTRAN program has led us to the surrogate objective of minimizing the average amount of work-in-process mail at each dispatch time.? Ostensibly, the amount of work-in-pr~ess mail is directfy related to flow time of the average piece of mail in the system. Other objectives are stated in the form of constraints. The maximum Sunday premium hours allowed is 4800 hours and the maximum night premium is 20,~ hours. Solutions which result in assigned premium hours less than these upper bounds are obviously preferable to those that are at the maximum values. In addition, the objective of maintaining acceptable administrative costs is reflected in the defini~on of the set of aIlowable tour start time, lf = {12 midni~t, 4 a.m., 8a.m., 12 noon, 4p.m., 8p.m.).
In step 5 of the heuristic procedure discussed above it was stated that one of the K top ranked tour windows would be randomly chosen for the assignment of the next group of personnel. Having more than one choice for the assignment at each iteration allows the heuristic procedure to generate more than one solution to the problem. In this way tradeoffs between the primary objective and the sub-objectives can be evaluated. Our initial computations with the procedure focus on the effect of K in generating good alternative solutions to the problem. Table 1 contains the results of a number of experiments for values of K from one to ten. Only one replication is made for K = 1 since it generates a deterministic work force tour assignment solution by always choosing the top ranked tour window at each iteration. Values of K > 1 will yield a set of different work force tour assignment solutions: therefore, 16 replications are made. for K = 2,3, . . . , 10. For each K > 1 Table 1 identifies the average work-&process inventory, night manhours used, and Sunday man-hours used over the 16 replications. Table 1 also identifies the lowest work-in-process inventory level achieved within each set of replications along with the night and Sunday manhours needed to achieve that work-in-process inventory IeveI. *The authors wish to acknowledge the ge~rou~ assistance of the Columbus, Ohio Sectional Center Facility in gathering the data and information used in this study. Much of the data used in the study, such as work center capacities and productivity rates. are virtually identical to the reat situation. The post office is labefed hypothetical because certain data, such as the percentages of flow between stations and mail volume data, were not readily available and had to be approximated. &his post o&e has either a local or outgoing dispatch in 70 out of the 84 possible bi-hourly time intervals of the week. Since a high proportion of the time intervals have dispatches we have elected to set a:,,..= 1 for ail m, IV, q and n for these initial experiments.
M.J. SHOWALTER etal.
266
Table 1. Ex~rimental results Work-in-process inventory
K 1 2 3 4 5 6 7 8 9 10 Random
Average of replications 163,191 156,560 162,427 164,012 156,039 -169,383 167,701 169,948 171,298 173,637 730,076
Sunday manhours utilized
Night manhours utilized
Lowest
Average of replications
Used by low W.i.P.
Average of replications
Used by low W.i.P.
Number of replications
139,443 132,655 141,624 131,811 128,996 132,229 136,701 146,185 138,681 670,968
1,864 2,624 2,904 2,%8 3,236 3,304 3,608 3,632 3.6% 3.712 4.297
2,416 3,048 2,720 3,208 3,328 3,552 3,656 3,560 3,488 4,060
18,870 18,716 18,490 18,1% 18,078 17,852 17.7% 17,694 17,810 18,028 14,963
19,090 18,880 18,440 18,420 18,120 18,430 18,020 17,520 17,890 12,020
16 16 16 16 I6 16 16 16 16 3
I
The solution achieved for K = 1 attempts to match manpower assignments to the pattern of mail arrival because of the values of a”’ ,,.,q,n chosen for this study. Since a high proportion of the time periods in a week have dispatches, setting K = 1 could be expected to do well with respect to the dispatches. As K increases it should be noted that the “attainable” solution space also increases.* Interestingly, the attainable solution space for a given K’ is contained in the solution space attainable for any other value of K > I(‘. Thus, for example, the solution space for K = 10 contains the solution that was found for K = 1, however, the probability of generating the K = 1 solution when K = 10 is obviously quite small. Reference to Table 1 indicates that there are better solutions with respect to service (as measured by work-in-process inventory level) for values of K > 1 than was found for K = 1.t Only K = 2,3,5 were capable of generating an average of work-in-process inventory for all sixteen runs below the 163,191 pieces associated with K = 1. However, it is not the average performance of K > 1 values that is of importance, rather it is the potential of K > 1 values to generate solutions with superior service levels. From Table 1 it can be seen that the lowest work-in-process inventory level generated by K > 1 values, in each case, is superior to that generated by K = 1. As noted in Table 1 three experiment runs were made with a random heuristic which randomly generated a work force tour assignment solution. The average work-in-process inventory of 730,076 pieces for the RANDOM model indicates the superiority of a heuristic methodology which focuses on mail arrival volumes and system work flows. Assuming that each set of 16 replications are normally distributed Fig. 3 shows the average service level for values of K, two standard deviations around the average service level, and the highest service level achieved for each K value. Obviously, it is desirable to identify work force tour assignment solutions in the area of - 25 for K > 1, if service level is the primary criterion. Determination of the specific K > 1 value to utilize necessitates consideration of the tradeoff between solution quality and computational efficiency. Table 2 compares the proportion of replications for each K > 1 value that generated service levels exceeding that of K = 1 with the average computer processing time for the set of replications. It seems apparent that a larger proportion of improved solutions can be generated for (1 c K < 6) than when K > 6. The average compute time per experimental run tends to increase as K increases suggesting that a potentially greater number of improved solutions can be generated when (1~ K < 6) for a given amount of resources. It is not exactly. clear -what K value in the range (1 < K ~6) should be used; however, K = 2 did generate the largest proportion of improved solutions for the lowest average level of computational effort. *The “attainable” solution space is defined here as the subspace of final solutions which can be generated by applying the heuristic algorithm to a specific problem. tit is interesting to note that the capacitated solution step 2 of the heuristic yielded an average work-in-process inventory of 28,176pieces. Thus some measure can be made of the “cost” of adhering to the 8-hr day, 5 consecutive days per week constraints as well as the upper bounds on Sunday and night manhours.
Manpower allocation in U.S. postal facilities
261
230 r
6
r”
DO
I
I
I
I
I
I
I
I
I
I
I
2
3
4
5
6
7
6
9
IO
K
factcr level
Fig. 3. Service levels for K values.
Tablk 2. Service
K
level and efficiency
% of replications exceeding K = 1service level 68.75% 50.0% 68.75% 62.5% 31.25% 31.25% 37.5% 25.0% 25.0%
computational
Average CPU time utilized (set) 13.4 15.8 15.98 16.2 16.4 16.77 17.38 15.98 16.4
The capability of K > I to develop a variety of superior solutions allows management to assess tradeoff in costs and service. For example, the best solution for K = 2 with respect to the primary objective of work-in-process inventory is 139,443 pieces, however, the cost of this solution over that of K = 1 is an additional 552 Sunday manhours and 220 night man hours per week. Within the subset of K = 2 replications with service levels exceeding that of K = 1 additional service-cost tradeoffs can be explored. Such tradeoffs could not be evaluated by management if K = 1. The relationship between service level and costs for the K > 1 values is presented in Figs. 4 and 5. As the value of K increases, the number of Sunday manhours increases. In contrast, as the value of K increases the number of night manhours decreases. These results are consistent with the logic of the heuristic model. When K = 1 the heuristic closely matches manpower assignments with mai arrival volumes. Since mail arrival volumes are extremely large during the evening Monday-Friday it is expected that few Sunday manhours are required but a large number of night manhours will be required. As K increases beyond one the matching of manpower tour assignments with maif arrival volumes is relaxed. Thus, it would be expected that more Sunday manhours and fewer night manhours would be specified. This relationship would provide some guidance to the postal manager in selecting a K value if he desires to effectiveIy control certain premium pay costs.
M.J. S~OWALTER et al.
268 do-
38-36 34 -
Sunday h@iW
/
,‘..
3230--
/’
/’
I--
--
--x
t-r&Ours utilized for service level
Lcnvest Sunday achwed
manhours in replicaticms
usage
28 26 24 22 20 -
K
factor
level
Fig. 4. Sunday manhours utilized for K values
S’Night manhours utilized highest servtce level
I
2
3
4
5
6
K factor
7
0
9
IO
level
Fig. 5. Night manhours utilized for K values
CONCLUSION
The tour assignment problem for an SCF is a large combinatorial problem. This paper presents a building heuristic procedure for specifying the tour assignment for each employee in the complement. The primary objective is to minimize the work-in-process mail inventories at dispatch time while the sub-objectives of minimizing Sunday and night manhours are recognized as constraints. Anaiysis of computational experience indicates there is an advantage to adjust the solution space so that more than one feasible solution can be generated by the procedure. This flexibility appears to provide the postal manager with a variety of solutions with which to analyze cost and service tradeoffs when selecting a specific work force tour assignment sofution.
REFERENCES 1, L. D. Bodin, Towards a general model for manpower scheduling: Parts I and II. Urban and Policy Sciences Program, State University of New York, Stony Brook, New York (September. 1972). 2. G. B. Dantzig, A comment on Edie’s traffic delays at toll booths. L Ops. Res. Sot. Am., 2,339-341 (1954). 3. S. R. Fox, The mail processing simulation as a teaching aid. Proc. National American fnsliture of Decision Sciences, 4 (November, 1972). 148-149. 4. N. B. Heller, I. T. McEwen and W. W. Stenzel. Computerized scheduling of police manpower: Methods and conclusions. Report NI 720186, National Institute of Law enforcement and Criminal Justice, 1 (March, 19731.
Manpower
allocation in U.S. postal facilities
269
5. W. B. Henderson and W. L. Berry. Determining optimal shift schedules for telephone trafhc exchange operators. Workina Paner No. 507. Krannert Graduate School of industrial Adminis~ati~. Purdue Universitv IAoril. 1975). 6. L. J. K&jewski and L. P. Ritzman. Disaggregation in manufacturing and serviceorganizations: &rvey of problem structures and research furdings. Working Paper WPS 761, College of Admi~st~tive Science, The Ohio State University (January, 1976). 7. G. C. Micahel, A Review of heuristic pro~ammi~, &c&on Sciences, Vol. 3, (No. 3) (July, I972), ?4-IO& 8. A. Newell, Heuristic Programming-III-structured problems. Progmss in Operations Research, Vol. III, ed. J. S. Aronofsky. John Wiley and Sons New York (I%!& 9. L. P. Ritzman, L. J. Krajewski and hf. J. Showalter. The disaggregation of aggregate manpower plans. Mgmt. Sci. Vol. 22, (No. I I) (July, 1976); 1204-1214, IO. M. Rothstein, Scheduling manpower by mathematical programming. fad. Engag (April, 1972). 29-33. 11. M. J. Showalter, A methodology for manpower tour assignment within a U.S. Postal-Service processing center facility. Unpublished doctoral dissertation, the Ohio State University (1976). 12, U. S. Postal Service, Towards Postal Excellence: The Repoir of the P&dent’s Commission-of Postal Otganization, Government Printing Office Washington, D. C., June (1968).
PergamonPress. 1977
Printed in Greai Britrm
MANPOWER ALLOCATION IN U.S. POSTAL FACILITIES: A HEURISTIC APPROACH M. J. S~OWAL~R,* Quantitative
Analyses ~pa~ment
College of Business Administration Madison, Wisconsin, U.S.A.
University
of Wisconsin, Madison,
and L. J. KRMEwsKIIand L. P. Faculty of Management
RITZMANS
Sciences College of Administrative Sciences, The Ohio State University, Ohio 43210. U.S.A. (Received
May
Columbus,
1976)
Scope and purpose-The purpose of this article is to explore the importance of scheduling the manpower resource in a service organization. The ability to schedule manpower to meet variations in demand can have a significant impact on the level of service that an organization can provide. This article focuses upon how to generate efficient employee work schedules for the work force in a large mail sorting facility of the U.S. Postal Service. These facilities are highly labor inten&e and service to the public depends largely upon how quickly the work force can process the mail. These mail sorting facilities are multi-stage processing systems, therefore, the exact time an employee is scheduled to work and the processing stage at which he/she is assigned largely determines how rapidly mail moves through the system. An efficient rule of thumb procedure is outlined for assigning each employee in the work force. General applicability of this procedure is relatively limited due to the specialized nature of the constraints in a mail sorting facility. Abstract-The work force tour assignment problem in a U.S. Postal Service Sectional Center Facility consists of specifying the weekly tour start time, work center assignment, and mail class responsibility for each postal employee. Any efficient work force assignment must be made subject to a number of constraints including tour schedule requirements, work center capacities, mail arrival volumes, and mail flow patterns through the processing system. A heuristic aigo~thm of a building, or cons~ction, nature is described which evaluates these many constraints in developing efficient work force tour assignment solutions. Once the problem setting and the heuristic algorithm are described, the use of the procedure is demonstrated by analyzing a hypothetical problem. INTRODUCTION
The United States Postal Service (USPS) has been besieged with problems for years, primarily in the areas of customer service and deficit budgets. These problems have taken on added significance since the reorganization on 1 July, 1971. In striving to meet the higher expectations of the public[l2], postal management can act on two different levels. The first level includes more strategic alternatives such as modernization programs, mechanization, and the separation of mail processing into two subsystems fbuik and general). The second level is the tactical one of finding better ways to allocate limited resources so as to best meet service standards. This study deals with this second level of managerial discretion. More specifically, its goal is to develop a satisfactory meth~ology for making work force tour ussign~enfs in a sec~ionul center facility (SCF). *Michael J. Showalter is Assistant Professor of Quantitative analysis, College of Business Administration, University of Wisconsin-Madison. He received his Ph.D. in Production Management from The Ohio State University. He is author of papers appearing in Monngemenl Science and Business and Economic Review, as well as a number of regional and national conferences. He is a member of TIMS and AIDS. tLeRoy J. Krajewski is Professor of Management Sciences, College of Administrative Science at the Ohio State University. He received his Ph.D. in Production Management from the University of Wisconsin-Madison. He is the author of papers appearing in Management Science, Bell Journai of Economics and Management, DeciSion Sciences, and the Proceedings of the Fourth Annual Simulation Symposium, as well as in the proceedings of a number of regional and national conferences. He is a member of TIMS and AIDS. Starry P. Ritzman is a Professor on the Faculty of Management Sciences, College of Adminstrative Science at Tbe Ohio State University. He received a D.B.A. in Production Management from Michigan State University. His papers have appeared in ~anagemenr Science, &r&ion Sciences, and P~ceedings of the Fourth Annual Simulation Symposium, as well as the proceedings of several national conferences. He is a member of TIMS, ORSA, and AIDS. 257
258
M. J.
SHOWALTERet al.
THE SCF SYSTEM
There are approximately 600 sectional center facilities located throughout the United States. Each SCF, employing between 1000 and 2000 persons, has responsibility for mail delivery and service to a particular geographic region of the country. Nonetheless, all SCFs have a set of common characteristics which are important considerations in the tour assignment problem. Mail classifications
and arrival patterns
Mail processed at an SCF is classified into several categories, based upon its origin (incoming or originating), priority and destination (local or outgoing). For example, mail can be classified into 8 different categories upon its arrival: originating or incoming airmail, originating or incoming first-class mail, originating or incoming second-class mail, or originating or incoming third-class mail. Exiting mail can similarly be classified 8 ways. Furthermore, mail enters an SCF with sizable peaks and valleys in the arrival pattern, depending on the hour of day and day of the week. In addition, these time dependent arrival patterns differ by mail classification. Mail processing
and dispatch schedules
A typical SCF has 20 identifiable work centers for processing the different mail categories. Each of the work centers represents a different kind of processing operation. Mail arriving at an SCF for processing may pass through several of these work centers before it is ready to be dispatched. The routing of mail through the work centers is a function of the mail’s priority designation, physical characteristics, and destination. Figure 1 illustrates the routing of first-class mail destined for outgoing dispatch in a typical SCF. Similar subsystems of work stations can be defined for each of the other mail classifications. Each of the nodes in Fig. 1 represents a different work center. The number within each circle is an identifying code. The decimal value associated with the entering branches represents the proportion of mail arriving at a work center which is first-class mail. The decimal value associated with the remaining branches represents the proportion of mail leaving the source node which is transferred to the next work center. Figure 1 points out clearly that an SCF is a multistage processing system. For the subsystem of 8 work centers in Fig. 1, the first stage work centers are 010,020 and 180. Stage 2 work centers are 035,060, and 080. The last stage consists of 040 and 070. It should also be noted that some work centers process more than one mail class; i.e. they appear in more than one mail subsystem. Dispatch schedules
The dispatch, or shipment, of mail from an SCF occurs according to a daily schedule. There are dispatch schedules for both local and outgoing dispatches. Over the short run these dispatch times are fairly inflexible; this is particularly true for outgoing dispatches transported by independent contractors (truck, air taxi and commercial airlines). The dispatch schedule is an 1
0 u
Fig. 1. Outgoing first-class mail subsystem.
Manpower allocation in U.S. postal facilities
259
important determinant of the quality and reliability of service. No matter how quickly mail is processed, it cannot be shipped until the next regularly scheduled dispatch. In a similar fashion, mail not completely processed prior to a first dispatch time must await the second scheduled dispatch. Other parameters
Over the intermediate time horizon of one year, the operating capacity of an SCF is relatively fixed. The processing capacity C, of work center w is limited by the number of individual work stations available. With the exception of work center 080, manpower can be added to a work center during any time span until its processing capacity limit is reached.* Another parameter which is relatively fixed in the short run is the productivity that can be expected from an employee at each center. This productivity rate remains the same regardless of the mail class; the major determinant of productivity is the type of task performed. THE TOUR ASSIGNMENT
PROBLEM
The work force tour assignment problem within an SCF can be defined as the assignment of each employee in a total work force pool (or “complement”) to a specific work week (by hour and by day) to process a specific mail class at a specific work center. Solution space and constraints
Tour assignments are constrained by the national postal labor contract and characteristics of the SCF processing system as described in the previous section. Each career regular (full-time) employee is allowed 40 hr of work per week. His work week consists of 5 consecutive 8-hr working days. The daily shift is 9 hr in duration with 1 hr allowed for meal time. The daily shift start times for all 5 consecutive work days must be identical. Postal management has complete freedom within these constraints to specify an employee’s tour assignment. Thus, an employee may begin his 5-day tour at any time of the day on any day of the week. By simply specifying a tour start time, the entire work week is defined. For example, a tour start time of Tuesday at 12 a.m. means that an employee will work from 12a.m. to 9a.m. on Tuesday through Saturday of each week. The 40 working hours so specified by a tour start time are referred to in this paper as its tour window. For the purposes of this study, the set of tour start times is restricted to 12 2-hr periods during each day. Coupled with the constraints cited earlier, this reduces the number of alternative solutions for each employee to 6720 (84 tour start times x 20 work centers x 4 mail classes). However, the problem has been specified as one of “work force” manpower tour assignments. This view of the problem reveals the combinatorial nature of the solution space. A tour assignment must be specified simultaneously for a thousand or more employees. The tour assignment of one employee has an effect on the most desirable tour assignments for all other employees. This interdependence between employee tour assignments is traced to the interdependence of the work centers in the multi-stage processing system. Objectives
The paramount objective in SCF tour assignments is to maximize service or the throughput rate of mail, given the available complement size. Service can be measured as the total volume of mail making the closest dispatch time after its arrival to the system. In addition to this objective, there is also a second objective of minimizing the number of Sunday and night manhours called for by the workforce tour assignment. More specifically, a 25% pay premium must be paid for any 8-hr shift which requires an employee to work during any portion of the time interval from 12 p.m. Saturday to 12 p.m. Sunday. Furthermore, a 10% premium is paid for all hours worked between 6 p.m. and 6 a.m. every day of the week. A third objective of management may be to limit the number of allowed daily tour start times, due to the administrative costs and confusion associated with having too many tour start times per day. *Work center 080 is mechanized with several employees manning each machine. It is highly desirable to assign a full crew to a*machine.if it is to be operated during a tour. CAOR
Vol. 4 No. 4-C
260
M.J.SHOWALTER
et al.
ALTBRNATESOLUTIONPROCEDURES
Several procedures have been reported in the literature for shift scheduling in service organizations, ranging from heuristic algorithms to optimi~tion techniques[l,2,~, IO]. Unfortunately, these procedures are designed for single-stage operations rather than the multistage SCF environment. Other reasons why these approaches are inadequate for an SCF are (1) the existence of dispatch times and (2) the existence of several objectives. The most prominent solution methodologies to consider for such a multi-stage problem are (1) optimiza~on techniques such as integer or linear programming, (2) simulation, and (3) heuristic pro~amming. Optimization techniques It is theoretically possible to represent the SCF work force tour assignment problem with a gigantic integer programming model. The objective function would be to minimize system inventory levels over the planning horizon. However, even just a one-day planning horizon results in a matrix size of 4000 by 8000. Extending the model to accommodate the necessary seven-day horizon makes such an optimization approach computationally unattractive and unwieldy. Another shortcoming is the necessity of a single objective function. Simulution One possible non-optimizing technique which is useful for evaluating complex multi-sage systems is simulation. In fact, this approach has already been developed for the SCF tour assignment problem[3]. This simulation application is based on a man-machine interaction in which the quality of solution obtained is a function of the parameter values defined by the decision maker. It allows the evaluation of tour assignments generated through trial and error. Its major limitation is its reliance on the genius of the decision maker to supply good solutions. Heuristic programming Heuristic programming is a non-optimizing methodology which offers the advantage of generating good solutions to problems. The heuristic programming approach is particularly well suited for application to illstructured problems such as tour assignment [7,8]. Rather than using trial-~d~rror procedures for generating solutions, heuristic pro~mming searches a limited set of alternatives through application of certain decision rules to develop a good solution to the problem. For the above reasons, and the inherent flexibility of heuristic programming, this methodology, coupled with simulation for evaluation purposes, has been developed for solving the SCF manpower tour assignment problem. It is described in the next section. THE HEURISTIC ALGORITHM
This algorithm can best be characterized as a “building” (or construction) heuristic procedure which makes tour assignments sequentially for more and more employees until the full complement has been assigned. The objective is to obtain a good solution without explicitly considering the existing manpower tour assignment used by the SCF. The procedure focuses upon mail arrival volumes and their timing to make tour assignments to the first-stage work centers. These assignments, coupled with routing patterns and productivities, are then used to determine the arrival volumes to the second-stage work centers, and so forth. The exact procedure is best described as consisting of the series of steps’in Fig. 2. Each of the steps is elaborated in the following sections. Specify user and system inputs-step 1 Parameter specifications can be classified as either con~ollable or uncontrollable by postal management. Certain data inputs reflect the characteristics of the processing system while other parameters may be specified by management according to their policy decisions. System parameter inputs include: (1) maximum capacity of each work center, (2) productivity rates at each work center, (3) mail arrival patterns by mail category, (4) dispatch schedules by mail category, and (5) routing patterns (see Fig. 1) for each mail category. ControlIabie p~ameter inputs subject to management prerogative include: (1) total work force size (complements for the SCF, (2) maximum allowable Sunday manhours, (3) maximum allowable night manhours,
Manpower
261
allocation in U.S. postal facilities
STEP 111
i STEP X2
SIMULATE CAPACITY TOUR ASSIGNMENT
1 INITIALIZE PROCESSING STAGE (S=l) 1
I INCRWENT STAGE (s-s+l)
-
STEP f/7
I
MARE FINAL ADJUSTMENTS
INCREMENT MAIL CLASS (m=m+l)
I
6
OUT
Fig. 2. Building heuristic program.
and (4) allowed tour start times. The last input allows management to keep adminstrative costs within limit by restricting the variety of tour start times.
Simulate capacitated
tour assignment-step 2 The second step of the algorithm is a necessary precursor to the third step which allocates the total work force complement to individual work centers. The second step begins by generating a full (or capacitated) manpower allocation by which manpower is assigned to each work center w up to its capacity limit C, for each 2-hr time period during the entire week ignoring all other constraints. Given this manpower capacitated solution, a deterministic simulation of mail flows is executed using a simulation program called DEMPSI-CAP. The output of this simulation program is the “ideal” quantity of each class of mail which makes each dispatch during the week. More importantly for the next step, it determines the ideal values for L$j, where: Lt.,,=
number of manhours of labor productivity used by the capacitated solution to process class m mail at work center w during bi-hourly time period i on day j.
Allocate complement
to work center crews-step
3
The third step is to divide the total work force complement into “work center crews”. A work center crew is defined here as the number of employees assigned to process a particular class of mail sometime during the week at a specific work center. This apportionment is begun
262
M.J.
SHOWALTER d al.
by first converting the work force complement into work center groups G,:
W6W
(1)
where: Q = work force complement size, W’ = set of work centers to which a work center group, G,, has not yet been assigned, W” = set of work centers to which a work center group, G,, has been assigned, W = W’UW”; set of all work centers, M = set of all mail classes, C, = manpower capacity of work center w. The apportionment is completed by further allocating the work center group among the various classes of mail. Thus, each work center crew, &“, is specified as follows:
where: 84’ = set of classes of mail processed at work center w for which a work center crew allocation has not yet been made, M” = set of classes of mail processed at work center w for which a work center crew allocation has been made, M = M’ U M”; set of all classes of mail. This proportional allocation process effectively reduces the manpower tour assignment problem to one of determining the tour start times for each member in each work center crew. Rank feasible fours-step 4 Steps 4 and 6 are closely interrelated. They are performed iteratively to select tours for each employee in each work center crew 6,“. With these 3 steps, the heuristic first makes a tour assignment to each first-stage work center which must process the airmail (m = 1). If this fails to assign f?,” man-tours to each work center w which processes mail m, the 3 steps are repeated until the full work center crews are assigned.? When this is completed, it is possible (1) to simulate the mail inputs into the ~eco~~-~~ffgework centers which must process the airmail (m = 1) and then (2) to apply steps 3-5 in the same fashion to make tour assignments to these second-stage work centers. This process continues until all stages are completed for the airmail. The entire procedure is then replicated for each of the Iower priority classes of mail. The only purpose of step 4 is to rank the alternate tours for each work center processing mail m in the stage being considered. The rank is essentially based on the amount of unprocessed mail, summed over all time periods in the tour window being ranked. More specifically, the rank of the tour beginning in period i* on day j at a first-stage work center is based on the value of Vci*,j where:$ tAs discussed in step 7, it may not be possible to allocate &“’ fully to all work centers. In such cases, step 7 makes the necessary final adjus~nts. $The asterisk superscript on the i indicates that the time period i is in the set of allowed tour start times.
Manpower allocationin U.S. postal facilities
263
mcFM WQWlrn i*cZ, j=1,2,...,7
(3)
and: V$*,j = total volume of unprocessed class m mail at work center w over the tour window of
the tour beginning in period i* on day j, A& = volume of class m mail arriving at the SCF in period q on day n, D”,,.q,m= number of employees assigned to process mail class m at work center w during period q and day n (due to tour assignments made at step 5 during prior iterations), I?! = weighting factor, between zero and one, which specified the importance of mail class aW,,,n m, at work center w, in period q of day n relative to its next available dispatch time,? W&VI= ELJX!,,.; output of class m mail from work center w over a feasible tour, ti+,j,which begins on interval i* on day j, Fr = proportion of class m mail arriving at the SCF which goes directly to work center w for processing, H;, = bi-hourly productivity rate per employee assigned to work center w, Z, = set of allowed tour start times, Pi = set of five consecutive days beginning with day j, Qi*= set of four consecutive two-hour time periods beginning with period i*, and WIm = set of first-stage work centers which process mail class m (a subset of w). In a similar fashion, the ranking of tour windows for work centers in stage s(s > I) is performed by replacing equation 3 with:
WEW,” i*eZf j=12 , Y-*-Y 7 s = 1,2, *. . ,ybm
(4)
where: = output of mail class m from work center I( in stage (s-t) during period 4 on day n, as ~Z?.PI determined by a simulation of mail flows given the tour assignments at stage (s-1). = set of work centers processing mail m in stage (s-Z) whose output flows (partially or ZJWrn wholly) to work center w for further processing. B,,w = proportion of class m mail flowing from work center u directly to work center W,and ybm= number of work center phases required to process class m mail for a type b dispatch. Make a tour assignment--step 5 For a given mail class m and stage s, step 5 begins by randomly choosing a work center By not restricting this choice to the tour which has one of the highest K rankings, VEi*+jn tThis allows the user to pIace greater emphasison mail which is flowing into a work center at a time at (or before) its next scheduleddispatch.
264
M.J.
SHOWALTER eral.
window with the highest ranking (allowing K > l), the heuristic algorithm gains the desirable capability of generating a variety of good work force tour assignments. The number of employees assigned to the tour so selected for a first-stage work station is equal to:
( VZ,i*,jl(2O*li,. 1)> 0 m&f WE W*m
i*df j= 1,2, . * . , 7
(9
where: @ = remaining Sunday man-tour equivalents allowed but not yet assigned, and Y = remaining night man-tour equivalents allowed but not yet assigned. After computing an Nc’*.i value for a first-stage work center, step 5 concludes by updating the values of D&,n, Y, and Q, to reflect the new tour assignment. The procedure for making tour assignments to work centers in subsequent stages (s > I) is essentially the same, except that the set of work centers evaluated is mW;” (s = 2,. . . , ybm) rather than weW;“. Simulate maii flows-step 6 For a given class m mail, all tour assignments must be made for stage (s - I) work centers
before considering stage s work centers. After stage (s - I) assignments are completed, the mail flows must be simulated to determine the output rates of all stage s work centers processing class m mail. This output information on O&, which can be obtained by a partial simulation of SCF, is then used in equations 3 and 4 to rank tour windows and make tour assignments for stage 5. The information generated by these partial simulations can also be amalgamated after the heuristic algorithm has determined the complete work force tour assignment. This information is printed out, along with the final tour assignments, at the end of the run for evaluation purposes. Make jinaf a~j~s~rnen~s-step
7
It may occur that it is impossible for step 5 to fully allocate to work center w its full work center crew SC, given the constraints on allowed tour start times, manpower capacity, allowed night manhours, and allowed Sunday manhours. When this happens the program completely eliminates the heuristically generated manpower tour assignment for that work center. The program attempts to reallocate work center crews to three consecutive, non-overlapping tours so as to maximize the IeveI of manpower allocated to that work center. To determine which non-overlapping tours will be chosen; the value V,li*,j is again calculated using either equation 3 or 4 for each feasible tour window. After choosing the maximum V,li*.i value, a manpower assignment of Nci*,i is made using equation 5 for each mail class until the capacity constraint C, is reached or work center crew is exhausted. If the capacity constraint is reached, the procedure iterates to the next feasible, non-overlapping tour start time of period (i* + 4). The maximum volume of mail (max V$+& for the set of feasible tour windows is determined and manpower assignments of N $+Jj are made for each class of mail until the capacity constraint is reached or the work center crew is exhausted. To complete
Manpower
allocation in U.S. postal facilities
265
this procedure, the same approach is utilized for the remaining non-overlapping allowed tour start time of period fi* + 8) if any work center crew remains unassigned. This procedure of non-overi~pping tour assi~ment assures that each work center crew is fully assigned and a maximum allocation is made to that work center. THE
HYPOTHETICAL
POST
OFFICE*
The post office used in this study is modelled after an SCF in the Midwest which utilizes 20 different work centers (W) in its production design ranging from opening and culling stations to secondary sorting stations. The facility processes four classes of mail: airmail, first-, second- and third-class mail. A piece of mail may pass through anywhere from two to five processing stages (s) depending upon its class and ultimate destination, In addition, the productivity rates depend upon the work center and range from 640 pieces per bi-hourly interval at the oncoming preferential opening unit to 12,030 pieces per bi-hourly interval at the culling and canceiiing station. Given a deterministic mail arrivai pattern for each mail class which varies by day of the week and hour of the day, the problem is to find feasible tour assignments for each of 800 employees in the authorized complement such that certain objectives are achieved. The building heuristic has a built-in priority structure for the objectives by relegating ah but the main objective to the constraint set. The main objective to maximize service by maximizing the volume of mail of each class which makes its first available dispatch. The di~culty in monitoring the nearest dispatch for each arriving piece of mail in a FORTRAN program has led us to the surrogate objective of minimizing the average amount of work-in-process mail at each dispatch time.? Ostensibly, the amount of work-in-pr~ess mail is directfy related to flow time of the average piece of mail in the system. Other objectives are stated in the form of constraints. The maximum Sunday premium hours allowed is 4800 hours and the maximum night premium is 20,~ hours. Solutions which result in assigned premium hours less than these upper bounds are obviously preferable to those that are at the maximum values. In addition, the objective of maintaining acceptable administrative costs is reflected in the defini~on of the set of aIlowable tour start time, lf = {12 midni~t, 4 a.m., 8a.m., 12 noon, 4p.m., 8p.m.).
In step 5 of the heuristic procedure discussed above it was stated that one of the K top ranked tour windows would be randomly chosen for the assignment of the next group of personnel. Having more than one choice for the assignment at each iteration allows the heuristic procedure to generate more than one solution to the problem. In this way tradeoffs between the primary objective and the sub-objectives can be evaluated. Our initial computations with the procedure focus on the effect of K in generating good alternative solutions to the problem. Table 1 contains the results of a number of experiments for values of K from one to ten. Only one replication is made for K = 1 since it generates a deterministic work force tour assignment solution by always choosing the top ranked tour window at each iteration. Values of K > 1 will yield a set of different work force tour assignment solutions: therefore, 16 replications are made. for K = 2,3, . . . , 10. For each K > 1 Table 1 identifies the average work-&process inventory, night manhours used, and Sunday man-hours used over the 16 replications. Table 1 also identifies the lowest work-in-process inventory level achieved within each set of replications along with the night and Sunday manhours needed to achieve that work-in-process inventory IeveI. *The authors wish to acknowledge the ge~rou~ assistance of the Columbus, Ohio Sectional Center Facility in gathering the data and information used in this study. Much of the data used in the study, such as work center capacities and productivity rates. are virtually identical to the reat situation. The post office is labefed hypothetical because certain data, such as the percentages of flow between stations and mail volume data, were not readily available and had to be approximated. &his post o&e has either a local or outgoing dispatch in 70 out of the 84 possible bi-hourly time intervals of the week. Since a high proportion of the time intervals have dispatches we have elected to set a:,,..= 1 for ail m, IV, q and n for these initial experiments.
M.J. SHOWALTER etal.
266
Table 1. Ex~rimental results Work-in-process inventory
K 1 2 3 4 5 6 7 8 9 10 Random
Average of replications 163,191 156,560 162,427 164,012 156,039 -169,383 167,701 169,948 171,298 173,637 730,076
Sunday manhours utilized
Night manhours utilized
Lowest
Average of replications
Used by low W.i.P.
Average of replications
Used by low W.i.P.
Number of replications
139,443 132,655 141,624 131,811 128,996 132,229 136,701 146,185 138,681 670,968
1,864 2,624 2,904 2,%8 3,236 3,304 3,608 3,632 3.6% 3.712 4.297
2,416 3,048 2,720 3,208 3,328 3,552 3,656 3,560 3,488 4,060
18,870 18,716 18,490 18,1% 18,078 17,852 17.7% 17,694 17,810 18,028 14,963
19,090 18,880 18,440 18,420 18,120 18,430 18,020 17,520 17,890 12,020
16 16 16 16 I6 16 16 16 16 3
I
The solution achieved for K = 1 attempts to match manpower assignments to the pattern of mail arrival because of the values of a”’ ,,.,q,n chosen for this study. Since a high proportion of the time periods in a week have dispatches, setting K = 1 could be expected to do well with respect to the dispatches. As K increases it should be noted that the “attainable” solution space also increases.* Interestingly, the attainable solution space for a given K’ is contained in the solution space attainable for any other value of K > I(‘. Thus, for example, the solution space for K = 10 contains the solution that was found for K = 1, however, the probability of generating the K = 1 solution when K = 10 is obviously quite small. Reference to Table 1 indicates that there are better solutions with respect to service (as measured by work-in-process inventory level) for values of K > 1 than was found for K = 1.t Only K = 2,3,5 were capable of generating an average of work-in-process inventory for all sixteen runs below the 163,191 pieces associated with K = 1. However, it is not the average performance of K > 1 values that is of importance, rather it is the potential of K > 1 values to generate solutions with superior service levels. From Table 1 it can be seen that the lowest work-in-process inventory level generated by K > 1 values, in each case, is superior to that generated by K = 1. As noted in Table 1 three experiment runs were made with a random heuristic which randomly generated a work force tour assignment solution. The average work-in-process inventory of 730,076 pieces for the RANDOM model indicates the superiority of a heuristic methodology which focuses on mail arrival volumes and system work flows. Assuming that each set of 16 replications are normally distributed Fig. 3 shows the average service level for values of K, two standard deviations around the average service level, and the highest service level achieved for each K value. Obviously, it is desirable to identify work force tour assignment solutions in the area of - 25 for K > 1, if service level is the primary criterion. Determination of the specific K > 1 value to utilize necessitates consideration of the tradeoff between solution quality and computational efficiency. Table 2 compares the proportion of replications for each K > 1 value that generated service levels exceeding that of K = 1 with the average computer processing time for the set of replications. It seems apparent that a larger proportion of improved solutions can be generated for (1 c K < 6) than when K > 6. The average compute time per experimental run tends to increase as K increases suggesting that a potentially greater number of improved solutions can be generated when (1~ K < 6) for a given amount of resources. It is not exactly. clear -what K value in the range (1 < K ~6) should be used; however, K = 2 did generate the largest proportion of improved solutions for the lowest average level of computational effort. *The “attainable” solution space is defined here as the subspace of final solutions which can be generated by applying the heuristic algorithm to a specific problem. tit is interesting to note that the capacitated solution step 2 of the heuristic yielded an average work-in-process inventory of 28,176pieces. Thus some measure can be made of the “cost” of adhering to the 8-hr day, 5 consecutive days per week constraints as well as the upper bounds on Sunday and night manhours.
Manpower allocation in U.S. postal facilities
261
230 r
6
r”
DO
I
I
I
I
I
I
I
I
I
I
I
2
3
4
5
6
7
6
9
IO
K
factcr level
Fig. 3. Service levels for K values.
Tablk 2. Service
K
level and efficiency
% of replications exceeding K = 1service level 68.75% 50.0% 68.75% 62.5% 31.25% 31.25% 37.5% 25.0% 25.0%
computational
Average CPU time utilized (set) 13.4 15.8 15.98 16.2 16.4 16.77 17.38 15.98 16.4
The capability of K > I to develop a variety of superior solutions allows management to assess tradeoff in costs and service. For example, the best solution for K = 2 with respect to the primary objective of work-in-process inventory is 139,443 pieces, however, the cost of this solution over that of K = 1 is an additional 552 Sunday manhours and 220 night man hours per week. Within the subset of K = 2 replications with service levels exceeding that of K = 1 additional service-cost tradeoffs can be explored. Such tradeoffs could not be evaluated by management if K = 1. The relationship between service level and costs for the K > 1 values is presented in Figs. 4 and 5. As the value of K increases, the number of Sunday manhours increases. In contrast, as the value of K increases the number of night manhours decreases. These results are consistent with the logic of the heuristic model. When K = 1 the heuristic closely matches manpower assignments with mai arrival volumes. Since mail arrival volumes are extremely large during the evening Monday-Friday it is expected that few Sunday manhours are required but a large number of night manhours will be required. As K increases beyond one the matching of manpower tour assignments with maif arrival volumes is relaxed. Thus, it would be expected that more Sunday manhours and fewer night manhours would be specified. This relationship would provide some guidance to the postal manager in selecting a K value if he desires to effectiveIy control certain premium pay costs.
M.J. S~OWALTER et al.
268 do-
38-36 34 -
Sunday h@iW
/
,‘..
3230--
/’
/’
I--
--
--x
t-r&Ours utilized for service level
Lcnvest Sunday achwed
manhours in replicaticms
usage
28 26 24 22 20 -
K
factor
level
Fig. 4. Sunday manhours utilized for K values
S’Night manhours utilized highest servtce level
I
2
3
4
5
6
K factor
7
0
9
IO
level
Fig. 5. Night manhours utilized for K values
CONCLUSION
The tour assignment problem for an SCF is a large combinatorial problem. This paper presents a building heuristic procedure for specifying the tour assignment for each employee in the complement. The primary objective is to minimize the work-in-process mail inventories at dispatch time while the sub-objectives of minimizing Sunday and night manhours are recognized as constraints. Anaiysis of computational experience indicates there is an advantage to adjust the solution space so that more than one feasible solution can be generated by the procedure. This flexibility appears to provide the postal manager with a variety of solutions with which to analyze cost and service tradeoffs when selecting a specific work force tour assignment sofution.
REFERENCES 1, L. D. Bodin, Towards a general model for manpower scheduling: Parts I and II. Urban and Policy Sciences Program, State University of New York, Stony Brook, New York (September. 1972). 2. G. B. Dantzig, A comment on Edie’s traffic delays at toll booths. L Ops. Res. Sot. Am., 2,339-341 (1954). 3. S. R. Fox, The mail processing simulation as a teaching aid. Proc. National American fnsliture of Decision Sciences, 4 (November, 1972). 148-149. 4. N. B. Heller, I. T. McEwen and W. W. Stenzel. Computerized scheduling of police manpower: Methods and conclusions. Report NI 720186, National Institute of Law enforcement and Criminal Justice, 1 (March, 19731.
Manpower
allocation in U.S. postal facilities
269
5. W. B. Henderson and W. L. Berry. Determining optimal shift schedules for telephone trafhc exchange operators. Workina Paner No. 507. Krannert Graduate School of industrial Adminis~ati~. Purdue Universitv IAoril. 1975). 6. L. J. K&jewski and L. P. Ritzman. Disaggregation in manufacturing and serviceorganizations: &rvey of problem structures and research furdings. Working Paper WPS 761, College of Admi~st~tive Science, The Ohio State University (January, 1976). 7. G. C. Micahel, A Review of heuristic pro~ammi~, &c&on Sciences, Vol. 3, (No. 3) (July, I972), ?4-IO& 8. A. Newell, Heuristic Programming-III-structured problems. Progmss in Operations Research, Vol. III, ed. J. S. Aronofsky. John Wiley and Sons New York (I%!& 9. L. P. Ritzman, L. J. Krajewski and hf. J. Showalter. The disaggregation of aggregate manpower plans. Mgmt. Sci. Vol. 22, (No. I I) (July, 1976); 1204-1214, IO. M. Rothstein, Scheduling manpower by mathematical programming. fad. Engag (April, 1972). 29-33. 11. M. J. Showalter, A methodology for manpower tour assignment within a U.S. Postal-Service processing center facility. Unpublished doctoral dissertation, the Ohio State University (1976). 12, U. S. Postal Service, Towards Postal Excellence: The Repoir of the P&dent’s Commission-of Postal Otganization, Government Printing Office Washington, D. C., June (1968).