AN EFFICIENT HEURISTIC FOR DISCONTINUOUS TOUR SCHEDULING PROBLEMS WITH A SINGLE SHIFT TYPE
Tony R. Johns, College of Business, 340 Still Hall, Clarion University, Clarion, PA 16214 (814) 393-2104, E-mail: [email protected]
ABSTRACT
A new approach for discontinuous labor tour scheduling problems with a single shift type and fixed break placement is presented in this paper. Unlike earlier heuristics this approach seeks to exploit the structure of this problem by recognizing that differing amounts of scheduling
flexibility exist during the day and capitalizing on this differing flexibility.
This new approach to the discontinuous labor scheduling problem, when incorporated into a labor scheduling heuristic, proved to be superior to comparable heuristics in terms of total labor hours scheduled and in CPU time when compared under an accepted set of test conditions. This approach is simple enough to easily implement manually for smaller scheduling problems.
INTRODUCTION
In recent years, many industrialized countries have seen their economies shift from industrial to service oriented economies. This shift to service economies has spurred interest in labor scheduling research. As a result, numerous scheduling methods have been developed and applied to scheduling problems in areas such as airlines [3] [16] [17] [18] [33] [51], banks [42] [44] [45], healthcare [38] [49] [50] [56], police [41] [54], post office [28], maintenance [21] [40], and telecommunications [4] [32] [39] [46], in an attempt to improve the efficiency with which labor is scheduled. These methods have included goal programming [8] [47], dynamic programming [24], network flow models [53], working subsets [9] [26] and implicit modeling [6] [10] [12] [56] formulations along with numerous heuristic methods. However, with few exceptions [13] [46], none of these approaches have attempted to exploit the structure of the discontinuous labor tour scheduling problem.
This paper presents a new approach that exploits the structure of the single shift length, fixed break, labor scheduling problem and tests that approach with a new construction heuristic. It builds on the work on [20] who used a similar approach in a more complex environment but who also used a much more computationally intensive approach. The results obtained from using this heuristic are superior to those found by previous heuristics under an accepted set of test
conditions taken from the literature.
This new heuristic has the added benefit of being simplistic enough to be implemented manually or readily coded into a computer spreadsheet for smaller problems. While managers often have to resort to computer programs to solver larger problems, the simplicity of this heuristic means that managers who either lack or are unable to access appropriate scheduling software can
achieve very good solutions to small problems without this software. Previously in the literature, the Bechtold-Showalter (BS) heuristic [13] was billed as a method which could be employed
manually. As the reader will subsequently see, we present a heuristic which is clearly superior to the BS heuristic in terms of performance.
The remainder of this paper is divided into six sections. Section two reviews the literature and previous heuristics that have been applied to the labor tour scheduling problem. The third section presents the logic of the new heuristic. Section four describes the environment in which this research was conducted. Section five presents the experimental results and section six our conclusions and implications for future research. Lastly, in Appendix 1 we present a complete example of the heuristic.
LITERATURE REVIEW
The labor tour scheduling problem [5] consists of determining both the daily shift schedule and days-off for each worker over a given time horizon that is typically one week in length. This problem may be separated into the continuous and discontinuous environments, where the continuous environment is assumed to operate twenty-four hours per day, while the
discontinuous environment is assumed to have a period on non-operation. Alfares [2] proposed a ten category classification scheme for tour scheduling approaches. From those ten, the three applicable to this paper are LP-based solution (LP), Construction/Improvement (CI), and Manual solution (Mn).
Dantzig first proposed using a heuristic method to round linear programming solutions to the labor scheduling problem into feasible integer solutions [23]. Since this initial suggestion, multiple heuristics have been developed in an attempt to achieve optimal or near-optimal
solutions. These methods may be grouped into linear programming based (LP) and construction (non based) methods with the LP based methods generally proving to be superior [11]. LP-based heuristics begin by obtaining the solution to a formulation of the labor tour scheduling problem such as the following:
Minimize: Z = Σ x j (1) jεM Subject to: Σ a ijx j r i for all i ε I (2) jεM x j 0 for all j ε J (3)
Where:
M = the set of allowable tours;
I = the set of operating periods used to represent the weekly planning period;
r i = the number of employees required to work in period i;
x j = the number of employees assigned to work schedule j;
a ij = 1 if period i is a work period in schedule j; 0 otherwise.
In this formulation, equation (1) minimizes the total number of labor hours scheduled and
equation (2) ensures that all demands for labor are met or exceeded (understaffing is not allowed) while (3) represents the non-negativity constraints. Dantzig [23] originally suggested rounding the fractional x j's into integers if the initial solution was not integer. However, due to constraint
(2) not allowing any understaffing, this can result in an infeasible solution thus necessitating the adding of additional employees to satisfy constraint (2). The principal difference then among the LP-based heuristics is the logic employed in arriving at a feasible integer solution.
Easton [25] described LP-based heuristics as a three phase process consisting of initialization, integerization, and augmentation. In the initialization phase, the LP solution is determined. If this solution is not integer, then the second or integerization phase is used to make the solution integer. This second phase either truncates [47] or rounds [32] [39] the LP solution into an integer solution. If the resulting solution is not feasible then augmentation or the adding of employees occurs until the solution becomes a feasible solution.
Construction heuristics address labor scheduling problems without requiring the use of a LP package. Beginning with no employee assignments, these heuristics add employees iteratively until all scheduling requirements are satisfied. The logic used in construction heuristics is often the basis for the creation of improvement heuristics. The primary difference being that
construction heuristics start with zero employment assignments and add employees while improvement heuristics seek to improve on existing solutions by adding and/or subtracting employees to produce better solutions.
In conjunction with both LP-based and construction heuristics, improvement routines have often been employed in an attempt to improve existing solutions [14] [32] [39] [14] [15] [19] [30] [27] [55] [43] [31].
Several heuristics presented as manual methods have been created for versions labor scheduling problems. Typically very restrictive simplifying assumptions such as only three starting times per day, no breaks, constant demand and a homogenous workforce were used for these heuristics [37] [34] [35] [36] [48] [1].
Of the previously published heuristics, only two attempted to exploit the structure of the
discontinuous labor scheduling problem. The first of these is the Two-Phase heuristic developed by McGinnis et. al. [46]. This heuristic first applies a shift allocation procedure to determine the shifts that are to be worked, and then applies a shift assignment procedure to group the shifts into tours.
The second method that seeks to exploit the structure of the discontinuous environment is the Bechtold-Showalter (BS) heuristic [13]. The BS heuristic is also a two phase heuristic which decomposes the tour scheduling problem into shift and days-off subproblems. In the first phase, a shift scheduling problem is solved for each day of the weekly planning horizon. The outputs of
the first phase are then used as a set of inputs for the second phase. In this second phase, a series of days-off problems are solved using a previously published [7] days-off algorithm.
While both of these methods sought to exploit the nature of the discontinuous labor tour scheduling problem, they both arrive at solutions by solving what are considered to be the component parts of the tour scheduling problem, which are the shift and days-off subproblems, sequentially. While this approach may yield satisfactory results, it ignores any possible
interaction that occurs between the shift and days-off subproblems when they are combined to form the tour scheduling problem. Thus, while very effective or optimal methods may exist for solving each of the subproblems, sequentially applying these methods to solve the tour
scheduling problem may result in solutions that are inferior to the optimal solution. Therefore, a heuristic that seeks to exploit the structure of the tour scheduling problem and not the shift and days-off subproblems may produce results that are superior to those of existing heuristics.
THE RS HEURISTIC
Most existing heuristics use a selection criterion for adding employees which selects the employee to be added from the set of all allowable tours. The new approach presented in this paper seeks to exploit the structure of the discontinuous labor tour scheduling problem by recognizing that, while a substantial portion of the allowable tours can be used to cover demand in the middle hours of the day, only a minimal number of tours are available to meet demand that occurs at the beginning or end of the day.
Specifically, for the first (last) hour of the day, there is a restricted set (RS) of only those tours which begin (end) in that hour with which to meet demand. The approach of the RS heuristic is to start by assigning employees to the tours which cover the demands at the beginning and end of the day, then to work to the middle of the day. When determining which days-off combination to use when assigning an employee to a tour, the heuristic chooses the tour which covers the highest total labor demands for the first hour of the tour. If there is a tie among competing tours, the tie is broken by selecting the tour, from the tied set, which covers the highest total labor demands from the second hour of the tour. This tie-breaking process is then repeated until the tie is broken or, if the last hour of the tour fails to break the tie then the first tour in the tied set is selected. After satisfying all of the labor demands for the first hour of every day, the RS heuristic moves to the last hour of the day and assigns employees to tours to satisfy all of the labor demands for the last hour of every day. When working with the last hour of the day, ties are broken by looking to the next earliest hour of the day. Once this has been completed, the heuristic then alternates from the earliest hour of the day with a zero labor demand to the latest hour of the day with a non-zero labor demand until all demands for labor have been satisfied. In Appendix 1 we present an example of the RS heuristic in a simple environment which allows us to work through a
complete example of the application of the heuristic.
EXPERIMENTAL ENVIRONMENT
To facilitate the comparison of the RS heuristic to previous heuristics, the operating environment and labor requirements used by Bechtold, et. al. [11] were used in this study. This environment
was chosen because it allowed for a clear and direct comparison of the new heuristic to previous heuristics.
This environment was assumed to operate 16 hours per day, 7 days per week, and was subject to hourly variations in demand. All employees used were full-time employees and were scheduled to work five days per week with the two days-off allowed to be either consecutive or
nonconsecutive. Each employee worked an eight-hour shift each day, during which there was a one-hour meal break. The meal break was centered in the shift, which resulted in the employee working four hours, taking a one-hour meal break, and then working the remaining four hours of the shift. Employees were allowed to be scheduled to begin work in any hour of the operating day as long as their shifts did not extend past the end of the operating day.
For each employee, shift start times were required to be the same for each day of the week
resulting in eight allowable start times for each employee. This resulted in a total of 168 possible tours (21 days-off combinations x 8 allowable start times) from which to assign employees. The three sets of demand patterns used in this study were characterized by a mean demand of 50 employees per hour and amplitudes of 8, 16, and 24, where amplitude is defined as the difference between the maximum and minimum demand across the day or across the week. Each set of demand patterns was generated using six shapes of demand. These six shapes were level, trend, concave, convex, sine, and bimodal. Each of these shapes was used to generate daily and weekly demand variation, resulting in 36 unique demand patterns. Since the level-level combination had an amplitude of zero, it was omitted from the set of patterns, resulting in 35 demand patterns. The combination of 35 patterns and 3 different amplitudes resulted in 105 test problems. Bechtold et. al. [11] identified nine heuristics in their examination of heuristic performance. Of these nine, four were LP-based heuristics and five were construction heuristics. The LP-based heuristics were the Linear Programming Adjustment (LPA), used by Dantzig [23] as well as by Krajewski, Ritzman, and McKenzie [42]; Henderson and Berry (HB) [32]; Keith (K) [39]; and Morris and Showalter (MS) [47] heuristics. Their analysis indicated the (K) and (MS) heuristics to be the best of the LP-based heuristics; consequently, these two heuristics were employed in this research for comparative purposes.
The five construction heuristics were the Buffa, Cosgrove, and Luce (BCL1 and BCL2) [22]; McGinnis, Culver, and Deane (MCD1 and MCD2) [46], and the Bechtold and Showalter (BS) [13] heuristics. Of these five, the BS heuristic was found to produce results that were
consistently superior to those of the other construction heuristics. Consequently, the BS heuristic was chosen to be one of the comparative heuristics used in this research.
The resulting experimental design (Table 1) was a full factorial design with two factors; heuristic method at four levels (K, MS, BS, RS) and amplitude of demand at three levels (8, 16, and 24), with 35 observations per cell. Data were collected for the (4x3x35) observations with respect to primary criteria of mean percent above the ILP global optimal solution, the number of solutions obtained that were equal to the global ILP optimal, and CPU time. Secondary criteria of the percentage of schedules having consecutive days-off, the total number of active tours, and the minimum number of daily start times were also employed to further compare the performance of the heuristics.
TABLE 1 Experimental Factors and Factor Levels
────────────────────────────────────────────────────────────── Factor A - Heuristic method
1. Keith (K)
2. Morris and Showalter (MS) 3. Bechtold and Showalter (BS) 4. Restricted Set (RS)
────────────────────────────────────────────────────────────── Factor B - Amplitude of Demand
1. 8 employees 2. 16 employees 3. 24 employees
────────────────────────────────────────────────────────────── COMPUTATIONAL RESULTS
The RS heuristic was coded in Visual Basic and implemented on a personal computer. The ILP optimal solutions for the 105 demand patterns were obtained using Lindo [52] on the same computer.
Primary Criteria
With respect to the mean percent above the global ILP optimal solution, the RS heuristic was superior to the other three heuristics. As shown in Table 2, the RS heuristic obtained the global ILP optimal for all of the demand patterns, while the K, MS, and BS heuristic averaged 0.12, 0.16, and 0.30%, respectively, above the global ILP optimal. With respect to the number of optimal solutions obtained, the RS heuristic obtained the optimal solution to all 105 problems, while the K, MS, and BS heuristic were only able to obtain 81, 71, and 52 out of a possible 105 optimal solutions respectively. While the RS heuristic found the optimal answers to all of the problems in this set, it is not an optimal heuristic. Due to break placement, the situation can arise where constructing an optimal solution requires assigning an employee to a start-time which has zero labor demand.
The run-time of the RS heuristic also compared favorably with those of the other three heuristics. The RS heuristic required 11 seconds to solve each of the three sets of 35 labor demands while its nearest competitor, the BS heuristic required and average of 15 seconds. The LP based heuristics, K and MS, required an average of 989 and 1243 seconds respectively. These results show the RS heuristic to be vastly superior to the LP-based heuristics, and superior to the BS heuristic in terms of run time.
TABLE 2 Primary Performance Criteria
────────────────────────────────────────────────────────────── Heuristic Method K MS BS RS
────────────────────────────────────────────────────────────── Mean % Above ILP Optimal
Amplitude 8 .11 .20 .38 .00
Amplitude 16 .18 .15 .29 .00
Amplitude 24 .07 .12 .22 .00
Average .12 .16 .30 .00
Number of Optimal Solutions Amplitude 8 27 21 14 35
Amplitude 16 24 24 20 35
Amplitude 24 30 26 18 35
Total 81 71 52 105
CPU Time (Seconds) Amplitude 8 972 1222 15 11 Amplitude 16 1001 1256 15 11 Amplitude 24 993 1250 15 11 Average 989 1243 15 11 ────────────────────────────────────────────────────────────── Secondary Criteria
The secondary criteria examined in this study, while useful in some contexts, may not constitute a fair comparison of the four heuristics. This is especially true for the RS heuristic due to its superior performance on the primary criteria. However, these secondary measures were included in this study to facilitate the demonstration of an additional feature of the RS heuristic.
With respect to the mean percentage of schedules having consecutive days-off, the RS heuristic outperformed the two LP-based heuristics at all three levels of amplitude by approximately 40%. The RS heuristic outperformed the competing construction based method (BS heuristic) by approximately 2%. Overall, the RS heuristic outperformed the other methods at all levels of demand amplitude with the exception of the BS heuristic at a demand amplitude of 24, where the BS heuristic obtained solutions with a minimum of 75% of the schedules having consecutive days-off, while the RS heuristic obtained solutions with an minimum of 73.96% of the schedules having consecutive days-off.
The performance of the RS with respect to this criterion can be attributed to the final tie breaking measure, which consists of selecting the first tour in the set of tied tours. The percentage of tours having consecutive days-off was increased by simply positioning the 56 tours which have
consecutive off first in the set of 168 allowable tours. Thus, if a tour with consecutive days-off had the same coverage statistic as a tour without consecutive days-days-off, the tie was broken in favor of the tour with consecutive days-off.
TABLE 3 Secondary Performance Criteria
Percent Of Schedules Having Consecutive Days-Off
Total Number of Active Tours
Minimum Number of Daily Shift Starts K MS BS RS K MS BS RS K MS BS RS Mean Amplitude 8 48.88 49.23 95.69 97.80 24.0 24.4 33.6 30.5 5.2 5.0 4.4 5.06 Amplitude 16 47.65 47.83 91.45 94.93 24.4 24.6 36.3 35.4 5.3 5.3 4.4 5.23 Amplitude 24 49.94 50.42 90.89 91.72 23.6 23.5 36.9 39.5 4.9 4.9 4.4 5.23 Standard Deviation Amplitude 8 14.29 14.26 4.02 1.78 4.3 6.1 6.1 5.2 1.2 1.2 1.1 1.72 Amplitude 16 10.22 10.17 6.96 3.75 5.3 6.2 7.8 8.4 1.2 1.2 1.1 1.76 Amplitude 24 11.19 11.17 6.75 6.45 4.8 4.2 8.0 11.7 1.3 1.3 1.1 1.76 Minimum Amplitude 8 13.45 14.29 87.02 94.88 15 15 21 21 3(6) 3(7) 3(11) 3(11) Amplitude 16 25.79 25.34 74.63 83.92 15 15 21 21 3(6) 3(5) 3(11) 3(11) Amplitude 24 26.20 26.96 75.00 73.96 18 18 21 21 3(6) 3(9) 3(11) 3(11) Maximum Amplitude 8 79.26 79.26 100.00 100.00 35 40 46 40 8(1) 8(1) 6(6) 8(6) Amplitude 16 71.43 72.35 100.00 100.00 39 44 52 52 8(3) 8(4) 6(6) 8(6) Amplitude 24 68.57 70.20 100.00 100.00 38 35 53 65 8(1) 8(4) 6(6) 8(6) * Numbers in parentheses indicate frequencies
The total number of active tours used by the construction heuristics (RS and BS) was noticeably higher than the number used by the LP-based heuristics. With a higher number of active tours generally considered to be undesirable due to the increased demands put on managers, this measure appears to favor the LP-based methods over the construction methods. However, with respect to the RS heuristic, it must be noted that it may be impossible to generate solutions equal to the ILP optimal without using a larger number of tours than those used by the LP-based heuristics which did not achieve all optimal solutions.
The same argument may be used for the minimum number of daily shift starts. The performance of the BS heuristic is better under this criterion than the performance of the other three methods. However, as was presented in Table 2, the performance of the BS heuristic for the primary criteria was the worst of the four methods. With respect to the RS heuristic, its performance is
approximately the same as that of the two LP-based heuristics. Thus, while the RS heuristic is not superior to the other heuristic methods using this criterion, it does not appear to be inferior.
CONCLUSIONS
The performance of the RS heuristic has validated the concept of exploiting the structure of the discontinuous labor tour scheduling problem. While previous attempts to exploit this structure have attempted to solve the shift and then the days-off sub problems sequentially, the RS heuristic exploited the structure of the entire problem. By selectively employing what is essentially a forward-looking days-off algorithm, the RS heuristic obtained solutions that were superior to those of existing heuristics. The RS heuristic is a days-off algorithm in nature in that it proceeds by selecting the best days-off combination at each iteration. However, the RS heuristic is different from preceding heuristics in that it is constrained to take an "ends-in" approach to generating a solution. The RS heuristic is forward-looking in that it breaks ties between days-off combinations by examining the requirements for labor in subsequent periods. Thus, the RS heuristic selects a days-off combination based not only on the initial period’s requirements for labor but on those of subsequent periods as well.
By focusing on the scheduling periods that have the least flexibility in terms of the number of possible tours that can be used to cover demand, the RS heuristic obtained solutions that were superior to those found by previous heuristics. For the LP-based heuristics, this suggests an alternative approach with respect to the rounding procedure used to generate the initial integer solution. Currently, many LP-based heuristics begin by rounding down the initial LP solution to obtain a starting integer solution. However, an alternative approach could be to round up or to round to the nearest integer the LP solution values that correspond to employee tours that begin (end) in the first (last) hour of the day.
A second potential advantage of exploiting the structure of the problem with respect to LP-based heuristics lies in the method in which LP-based heuristics arrive at a feasible integer solution from the initial integer solution. Currently, LP-based heuristics add employees to the initial solution which are selected from the set of all allowable employee tours. However, as demonstrated it is effective to use an "ends-in" approach when adding tours. Therefore, beginning by adding employees to those tours which cover demand in the beginning and end of the day may result in a superior solution.
The managerial implications of exploiting the structure of the discontinuous labor scheduling problem are obvious. Management is constantly seeking to improve operational efficiency by
improving the match between the supply of labor and the demand for labor. Either including logic that exploits the structure of the problem in current scheduling procedures, or implementing new procedures that are based on exploiting the structure of the problem can lead to substantial savings. Due to the simplicity of the logic necessary to exploit this structure, such procedures should be easily implemented with minimal cost.
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APPENDIX
Labor Demands and Tour Assignments Example Problem
To facilitate the reader’s understanding of the heuristic we present the following example of a five-day workweek which has consecutive days-off, and a twelve-hour operating day using five hour shifts.
As mentioned earlier, the RS heuristic starts with the labor demands in the first hour of the day, which are 1,2,3,3,3,2,1 as seen in panel 1. An examination of these demands reveals that a tour which covers the middle five of these days (which we have underlined for easy identification) is the tour which covers the highest total demand of any five-day period. This tour is shown in panel 2. One employee is then assigned to this tour which results in the remaining labor demands which are shown in panel 3.
Panel 1 Panel 2
Days of the Week Days of the Week
1 2 3 4 5 6 7 1 2 3 4 5 6 7 Hours of the Day 1 1 2 3 3 3 2 1 1 1 1 1 1 1 2 1 2 3 3 3 2 1 2 1 1 1 1 1 3 2 3 4 4 4 2 1 3 1 1 1 1 1 4 2 3 4 4 4 2 1 4 1 1 1 1 1 5 2 3 4 4 4 2 1 5 1 1 1 1 1 6 1 1 2 2 2 1 1 6 7 1 1 2 2 2 1 1 7 8 1 2 4 4 4 3 2 8 9 1 2 4 4 4 3 2 9 10 1 2 4 4 4 3 2 10 11 1 2 3 3 3 2 1 11 12 1 2 3 3 3 2 1 12
As we can see in panel 3, the remaining labor demands for hour 1 are 1,1,2,2,2,1,1. Given the five-day workweek in this example there are several tours which tie when the total labor demands are calculated for the days they cover. For instance, days one though five have a total demand of eight (1+1+2+2+2+2) while days two though six and days three through seven also have a total demand of eight. Therefore, the tie breaking rule of the heuristic is used which is to further evaluate the tied tours by examining the labor demands in the next hour. However, since the next hour, hour 2, has the same labor demand as hour 1 the tied set is unchanged. Therefore, we proceed to the third hour, which has labor demands of 2,2,3,3,3,1,1. As we can see, the tour that covers days one to five has the highest labor demands in hour three so it is the one that is selected. This tour is shown in panel 4.
Panel 3 Panel 4
Days of the Week Days of the Week
1 2 3 4 5 6 7 1 2 3 4 5 6 7 Hours of the Day 1 1 1 2 2 2 1 1 1 1 1 1 1 1 2 1 1 2 2 2 1 1 2 1 1 1 1 1 3 2 2 3 3 3 1 1 3 1 1 1 1 1 4 2 2 3 3 3 1 1 4 1 1 1 1 1 5 2 2 3 3 3 1 1 5 1 1 1 1 1 6 1 1 2 2 2 1 1 6 7 1 1 2 2 2 1 1 7 8 1 2 4 4 4 3 2 8 9 1 2 4 4 4 3 2 9 10 1 2 4 4 4 3 2 10 11 1 2 3 3 3 2 1 11 12 1 2 3 3 3 2 1 12
Panel 5 shows the remaining demand for labor and panel 6 shows the tour which was selected for the next employee assignment.
Panel 5 Panel 6
Days of the Week Days of the Week
1 2 3 4 5 6 7 1 2 3 4 5 6 7 Hours of the Day 1 0 0 1 1 1 1 1 1 1 1 1 1 1 2 0 0 1 1 1 1 1 2 1 1 1 1 1 3 1 1 2 2 2 1 1 3 1 1 1 1 1 4 1 1 2 2 2 1 1 4 1 1 1 1 1 5 1 1 2 2 2 1 1 5 1 1 1 1 1 6 1 1 2 2 2 1 1 6 7 1 1 2 2 2 1 1 7 8 1 2 4 4 4 3 2 8 9 1 2 4 4 4 3 2 9 10 1 2 4 4 4 3 2 10 11 1 2 3 3 3 2 1 11 12 1 2 3 3 3 2 1 12
As the reader can see in panel 7, all of the demands for labor in hour 1 are now zero, which indicates that it is time to move to the last hour of the day, hour 12.
Panel 7 Panel 8
Days of the Week Days of the Week
1 2 3 4 5 6 7 1 2 3 4 5 6 7 Hours of the Day 1 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 2 3 1 1 1 1 1 0 0 3 4 1 1 1 1 1 0 0 4 5 1 1 1 1 1 0 0 5 6 1 1 2 2 2 1 1 6 7 1 1 2 2 2 1 1 7 8 1 2 4 4 4 3 2 8 1 1 1 1 1 9 1 2 4 4 4 3 2 9 1 1 1 1 1 10 1 2 4 4 4 3 2 10 1 1 1 1 1 11 1 2 3 3 3 2 1 11 1 1 1 1 1 12 1 2 3 3 3 2 1 12 1 1 1 1 1
An examination of the demands for labor in hour 12 of panel 7 shows that the tour shown in panel 8 is the appropriate tour to use. Obviously, there are several tours which end in hour 12 which tie based on the selection criteria. These tours remain tied after using the eleventh hour as a tie-breaker however, the tenth hour breaks the tie and results in an employee being assigned to the tour seen in panel 8.
We will forgo explaining the remainder of the panels and instead note that the assignment of one employee to the tour in panel 16 results in the satisfaction of all labor demands in this example.
Panel 9 Panel 10
Days of the Week Days of the Week
1 2 3 4 5 6 7 1 2 3 4 5 6 7 Hours of the Day 1 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 2 3 1 1 1 1 1 0 0 3 4 1 1 1 1 1 0 0 4 5 1 1 1 1 1 0 0 5 6 1 1 2 2 2 1 1 6 7 1 1 2 2 2 1 1 7 8 1 1 3 3 3 2 2 8 1 1 1 1 1 9 1 1 3 3 3 2 2 9 1 1 1 1 1 10 1 1 3 3 3 2 2 10 1 1 1 1 1 11 1 1 2 2 2 1 1 11 1 1 1 1 1 12 1 1 2 2 2 1 1 12 1 1 1 1 1
Panel 11 Panel 12
Days of the Week Days of the Week
1 2 3 4 5 6 7 1 2 3 4 5 6 7 Hours of the Day 1 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 2 3 1 1 1 1 1 0 0 3 4 1 1 1 1 1 0 0 4 5 1 1 1 1 1 0 0 5 6 1 1 2 2 2 1 1 6 7 1 1 2 2 2 1 1 7 8 1 1 2 2 2 1 1 8 1 1 1 1 1 9 1 1 2 2 2 1 1 9 1 1 1 1 1 10 1 1 2 2 2 1 1 10 1 1 1 1 1 11 1 1 1 1 1 0 0 11 1 1 1 1 1 12 1 1 1 1 1 0 0 12 1 1 1 1 1 Panel 13 Panel 14
Days of the Week Days of the Week
Hours of the Day 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 2 3 1 1 1 1 1 0 0 3 1 1 1 1 1 4 1 1 1 1 1 0 0 4 1 1 1 1 1 5 1 1 1 1 1 0 0 5 1 1 1 1 1 6 1 1 2 2 2 1 1 6 1 1 1 1 1 7 1 1 2 2 2 1 1 7 1 1 1 1 1 8 0 0 1 1 1 1 1 8 9 0 0 1 1 1 1 1 9 10 0 0 1 1 1 1 1 10 11 0 0 0 0 0 0 0 11 12 0 0 0 0 0 0 0 12 Panel 15 Panel 16
Days of the Week Days of the Week
1 2 3 4 5 6 7 1 2 3 4 5 6 7 Hours of the Day 1 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 2 3 0 0 0 0 0 0 0 3 4 0 0 0 0 0 0 0 4 5 0 0 0 0 0 0 0 5 6 0 0 1 1 1 1 1 6 1 1 1 1 1 7 0 0 1 1 1 1 1 7 1 1 1 1 1 8 0 0 1 1 1 1 1 8 1 1 1 1 1 9 0 0 1 1 1 1 1 9 1 1 1 1 1 10 0 0 1 1 1 1 1 10 1 1 1 1 1 11 0 0 0 0 0 0 0 11 12 0 0 0 0 0 0 0 12