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Chapter 7: Numerical Analysis
7.1. Introduction In the previous chapters, the Insertion Algorithm (IA) and Two-stage Heuristic (TSH) were shown to be effective and efficient in solving the manpower-scheduling problem as demonstrated through the use of real data from the company. Both heuristics also generated solutions quickly. In this study, IA is developed to solve the problem without considering workforce synchronisation; meanwhile TSH includes this constraint. Since TSH can actually be used to solve both versions of model, it will be interesting to know the best algorithm under different settings. In this chapter, both heuristic solutions, the IA and the TSH, are evaluated further and compared with a range of data sets with different factors. (CPLEX is not included in this comparison analysis because of its limitation in solving the scheduling problem within a reasonable computing time.) All experiments were carried out on a Dell Pentium IV 1.8GHz computer. 7.2. Case study The same set of real data described in previous chapters was used to compare the performance of proposed heuristic solutions. On average, there were 140 to 160 flights (230-280 split aircraft maintenance tasks) each day. The company assigned 75 to 79 maintenance teams to load all aircraft on any given day, as shown in Figure 7.1. From Figure 7.1, both heuristic solutions outperform the manual solution developed by the company’s expert planner. All test data can be executed in less than 3 seconds. The IA reduces the required manpower by 10.39% (40 teams in total) on average. On the other hand, two different assumptions are considered by the TSH. Heuristic B takes into account the synchronisation of loading teams, while heuristic A allows different teams to carry out the operation any time in a given time window. In total, heuristic A manages to reduce the manpower by as many as 75 teams (19.59%) over the 5 day period. Even though synchronisation of teams is applied, heuristic B performs better than the IA (3.2% or 11 teams) and the real roster (13.33% or 51 teams).
Figure 7.1: Comparison of number of teams required by real roster and heuristic methods This comparative results shows that both heuristic solutions are able to solve the problem effectively and efficiently, even in instances with large problems. In the meal break sensivity analysis, loading teams are not given one hour meal break. The teams not only perfrom an extra hour of work but also save the travelling time to and from the service centre for the break. Indirectly, manpower productivity can also be increased due to time saved, including the unaviodable idle time caused by its time window constraint. The meal break sensivity results of both heuristic solutions are shown in Figure 7.2. On average, the differences are 6.69% (23 teams) for the IA, compared to over 8.41% (26 teams) and 8.7% (29 teams) for the TSH under different settings. The maximum difference is 13.24% (9 teams) on Day 4 by the TSH (A). This sensitivity analysis shows that both heuristic methods are capable of assigning and scheduling the aircraft efficiently, regardless of whether or not the meal break allocation is applied.
7.3.1 Description of the Test Problems We generated sixteen sets of problems, each consisting of ten test problems. These test problems were designed to highlight several demand factors that can affect the behaviour of routing and scheduling heuristics. These factors include demand distribution, demand size and the tightness of time windows. Demand distribution Two types of demand distributions were generated for performance analysis: demand with peak demand (P) and demand without significant peak (N). These reflect the norm of peak and off-peak seasons in real life situation. The aircraft arrival and loading times are randomly generated. Arrival times fall within a daytime period, while loading times vary, in range 1 to 5 units of time, as in the case study. Tightness of time windows In the case study, time windows are defined as aircraft arrival and departure times. Tightness of time window is defined as the ratio of the time window width to the processing time. Again, two types of time window tightness are included in the test problems: tight-width time windows (A) and a mixture of tight- and wide-width time windows (B). In test problem type A, the ratio of an aircraft’s time window width to its processing time is set to be 1 to 3. Meanwhile, in test problem type B, the ratio is 1 to 10. We randomly generate a time window’s ratio over a predefined interval. Given the aircraft’s arrival time, we then generate departure time by adding the time window width to the arrival time. Demand size Test problems that evaluate each demand distribution and time window are prepared in four different sizes: 100 aircraft, 250 aircraft, 500 aircraft and 750 aircraft. This is to test the robustness and efficiency of the proposed heuristic solutions. 7.3.2. Managerial Settings Each test problem generated is tested with different managerial settings, such as meal-break duration, length of working-shift and trip travelling limit. The managerial settings are: Meal-break duration: one-hour meal break, thirty-minute meal break, fifteen- minute meal break or no meal break Length of working-shift: four-hour shift, six-hour shift or eight-hour shift.
Trip-travelling limit: tight trip-travelling limit (loading, at most, three aircraft and two hours’ travelling duration) or loose trip-travelling limit (loading, at most, four aircraft and three hours’ travelling duration). 7.4. Computational Results To analyse the performance of the IA and TSH, we programmed them in C# and performed computational tests on the problem classes described in section 3.1. Solution quality is measured in terms of the minimum number of servicing teams and the computational time required to produce this solution. Table 7.1: Comparison of heuristic solutions to test problems Shift 4 hours 6 hours 8 hours Meal break Nil 15 mins Nil 15 mins
mins Nil 15 mins
mins
hour Trip constraint T L T L T L T L T L T L T L T L T L AP_100 I - I 2 - I - I - I - 2 I - I - I I AN_100 I - I 2 I I I I I I - 2 I - I - I I BP_100 I - I 2 - I I I I I - - I I I I I I BN_100 I - I 2 - I I I I I - 2 I - I I I I AP_250 I I I 2 I I I I I I - - I I I I I I AN_250 I^ -^ I^2 -^ I^ -^ I^ -^ I^ -^2 I^ -^ I^ -^ I^ - BP_250 I^ I^ I^2 2 I^ I^ I^ I^ I^ -^2 I^ I^ I^ I^ I^ I BN_250 I^ -^ I^2 2 I^ -^ I^2 I^2 2 I^ I^ I^ I^ I^ I AP_500 I^2 I^2 I^ I^ I^ I^ I^ I^ -^ -^ I^ I^ I^ I^ I^ I AN_500 I 2 I 2 I I I I 2 I - - I I I I I I BP_500 I I I 2 2 I I I - I 2 2 I I I I I I BN_500 I I I 2 2 I - I 2 I 2 2 I I I I I - AP_750 I I I I I - I I I I - - I I I I I I AN_750 I I I I - I I I I I - - I I I I I I BP_750 I I I 2 2 I I I - I 2 2 I I I I I I BN_750 I I I 2 2 I 2 I 2 I 2 2 I I I I I -
- I – Insertion Algorithm; 2 – Two-stage Scheduling heuristic
7.4.1. Analysis on Demand Size The number of test problem for which each heuristic obtains the best solution is plotted in Figures 7.3 and 7.4. In the case of demand size, the IA dominates the TSH in 14 out of 18 settings as shown in Figures 7.3 and 7.4. Overall, there is no general trend related to demand size for both heuristics. For the tight trip travelling settings, described in Figure 3, IA performs better than TSH in 7 out of the 9 settings. IA produces better solution than TSH in all four- hour working shift test problem. Similarly, IA performs better in both six-hour and eight-hour working shift when meal break allocation is applied. For TSH, there is no identified trend across different demand sizes, with respect to the number of superior solutions provided. However, IA shows a slight drop in number of test problems when the demand size increases. Similarly to the case of tight trip travelling settings, IA performs better than TSH in 7 out of 9 settings under loose trip travelling settings. However, the two settings for which TSH performs better are not the same settings as in tight trip travelling. IA produces better solution in all test problems in six-hour working shift and eight-working shift (when the meal break is considered). In summary, TSH performs better when six-hour or eight-hour working shifts are considered, no meal break is given, and under a tight travelling assumption. Furthermore, it is also performs better, in a loose travelling setting, with a four-hour working shift with a 15 minute meal break and eight-hour working shift with no meal break. 7.4.2. Analysis on Demand Distribution Next, we focus on the pattern of demand: demand with peak periods (P) and without peak periods (N) as shown in Figure 7. 5. The number of test problems for which the different heuristics dominate is reported. IA dominates the TSH in 29 out of 36 settings, 16 times under a setting with peak demand and 13 times where there is no peak demand. Furthermore, IA performs betters than TSH in all test problems, where tight trip travelling is set in peak demand distribution. When considering IA on its own, it performs better in the first setting (15 out of 18 settings) compared to latter setting (8 out of 18 settings), regardless of whether one considers tight or loose travelling settings. TSH appears to perform better in demand without peak.
7.4.3. Analysis on Demand Time Windows The results of the analysis of the effect of the time window size on the efficiency of the heuristics are similar to those obtained in the previous subsection. IA outperforms TSH under most of the settings (29 out of 36 settings) and has a better performance with tight time windows (14 out of 18 settings). To conclude, we can summarise the performance of both heuristics under different demand patterns, as shown in Table 7. 3 below. Table 7. 3 : Performance analysis of IA and TSH IA TSH Data distribution Performs better with data distributions with peaks (P) Performs better with data distributions without peaks (N) Tightness of time windows Performs better with tight time windows (A) Almost similar performance in both tight (A) and mixture of tight and wide time windows (B) Sample sizes Almost similar performance in all sample sizes Almost similar performance in all sample sizes 7.4.4. Analysis on Travelling Trip Setting A scenario based on the case study is simulated by setting the following parameters: an eight-hour working shift, a one hour meal break between the 3rd^ and 5th hour of working and tight trip travelling limit. The average efficiency gap of both heuristics is shown in Table 7.4 (under column of tight trip travelling), by (TSH - IA)*100/IA. In the tight trip travelling setting, TSH requires more servicing teams than the IA to complete the workload, especially in BP and BN test problems. On the other hand, the TSH performs slightly better with a sample size of 750 aircraft in all data sets. The gap in both AP-750 and AN-750 test problems are not only small in percentage but also two or more times less than the other smaller sample size test problems.
Figure 7.4: Demand size analysis on loose trip travelling setting
Figure 7.5: Time windows analysis Demand distribution with peak (P) Tight trip travelling Loose trip travelling 4H_0 4H_1 6H_0 6H_1 6H_2 8H_0 8H_1 8H_2 8H_4 4H_0 4H_1 6H_0 6H_1 6H_2 8H_0 8H_1 8H_2 8H_ IA 80 80 45 73 68 50 78 78 80 64 14 77 80 79 43 75 74 74 TSH 0 0 41 11 16 38 2 2 0 35 69 4 0 1 48 7 12 11 Demand distribution without peak (N) Tight trip travelling Loose trip travelling 4H_0 4H_1 6H_0 6H_1 6H_2 8H_0 8H_1 8H_2 8H_4 4H_0 4H_1 6H_0 6H_1 6H_2 8H_0 8H_1 8H_2 8H_ IA 80 80 39 56 42 34 80 80 80 60 11 77 80 80 22 67 71 68 TSH 0 0 48 34 45 55 2 2 1 37 69 4 1 0 64 22 17 19 Insertion Algorithm Tight trip travelling Loose trip travelling 4H_0 4H_1 6H_0 6H_1 6H_2 8H_0 8H_1 8H_2 8H_4 4H_0 4H_1 6H_0 6H_1 6H_2 8H_0 8H_1 8H_2 8H_ P 80 80 45 73 68 50 78 78 80 64 14 77 80 79 43 75 74 74 N 80 80 39 56 42 34 80 80 80 60 11 77 80 80 22 67 71 68 Two-stage Scheduling Heuristic Tight trip travelling Loose trip travelling 4H_0 4H_1 6H_0 6H_1 6H_2 8H_0 8H_1 8H_2 8H_4 4H_0 4H_1 6H_0 6H_1 6H_2 8H_0 8H_1 8H_2 8H_ P 0 0 41 11 16 38 2 2 0 35 69 4 0 1 48 7 12 11 N 0 0 48 34 45 55 2 2 1 37 69 4 1 0 64 22 17 19
7.4.5. Analysis on Meal Break Allocation We assume a similar setting to that considered in previous sections except for the fact that a meal break is not given to the loading team. In this sensitivity study, we focus on the scenario with the eight-hour working shift. Most of the computational results show the reduction statistic is similar in pattern to that obtained for the analysis which considers the meal break. This is demonstrated in Figure 7. 7 and Figure 7. 8. As expected, the number of loading teams required is less than that required under the scenario where the meal break is an must be scheduled during the working shift, regardless of the sample size. This is due to the fact that the loading teams have extra time from not having a meal break and the resulting unavoidable idle time. The comparative results for percentage reduction are given in Table 7.5. The improvement for the TSHs is significantly larger than that of the IAs. Furthermore, the manpower reduction is also greater than expected in the TSH, when the saved hours (missing the meal break) are divided by the working shift hours, for example a 28.24% (40 teams) reduction on the BN-tight trip for 750 aircraft compared to the expected 12.5% (18 teams). It shows that the TSH is able to solve the scheduling problem effectively even when the meal break is not applied. Figure 7. 7 : Computational result of the IA on AP test problems
Figure 7. 8 : Computational result of TSH on AP test problems On the other hand, the IA has the opposite computational result in the BN and BP test problems. The number of loading teams required is similar, regardless of whether a meal break is allocated within the working shift. Furthermore, more than half of the results show that even more loading teams are required when the meal break is not considered. This explains the time window cost evaluation component (as in equation (1) in Chapter 5) in the IA, where it tends to opt for the aircraft with the widest time window in order to insert more aircraft in the following iteration. By solving the BN and BP test problems, this evaluation component performs very well at the beginning. Later, though, the scheduling process is restricted with most of the aircraft with tight time windows. As the meal break functions as a divider of a shift, the situation worsens if the meal break is not applied. This happens especially in a large sample size with a short planning horizon. However, a time window cost evaluation component suits the following scenarios: tight trip travelling aircraft with tight time windows smaller sample sizes with a mixture of tight and wide time windows
Table 7. 6 : Working shifts analysis with meal break allocation (Four-hour shift ratio base) Tight trip travelling Loose trip travelling IA TSH IA TSH AP- 100 & 250
4H < 6H’ < 8H’ 6H’ < 4H < 8H’ 6H’ < 4H < 8H’ 4H < 8H’ < 6H’
AP-
4H < 6H’ < 8H’ 6H’ < 4H < 8H’
6H’ < 4H < 8H’ 4H < 6H’ < 8H’
AN-
4H < 6H’ < 8H’ 6H’ < 4H < 8H’ 6H’ < 4H < 8H’ 4H < 8H’ < 6H’
AN-
6H’ < 4H < 8H’ 6H’ < 8H’ < 4H
6H’ < 8H’ < 4H 6H’ < 8H’ < 4H
BP-
4H < 6H’ < 8H’ 8H’ < 4H < 6H’ 6H’ < 8H’ < 4H
4H < 8H’ < 6H’
BP-
6H’ < 8H’ < 4H 6H’ < 8H’ < 4H
6H’ < 8H’ < 4H
4H < 8H’ < 6H’
BN-
6H’ < 8H’ < 4H 6H’ < 8H’ < 4H
6H’ < 8H’ < 4H
4H < 8H’ < 6H’
BN-
8H’ < 6H’ < 4H 6H’ < 8H’ < 4H 6H’ < 8H’ < 4H
4H < 8H’ < 6H’
Table 7. 6 demonstrates that 6H’ dominates 4H as well as 8H’ in 19 out of 32 settings. Furthermore, 8 out of 19 leading results of 6H’ achieve manpower reduction more than 10% of the teams’ ration of 4H. Most of these excellent results are found in the loose trip travelling scenario for the IA (especially BN and BP data sets) and the tight trip travelling scenario for the TSH (especially large problem sets). However, 6H’ requires more loading teams in the loose trip travelling scenario for the TSH in small problem sets, where the team ratio can be as much as 1.37 - 1.41 of 4H. Even though 6H’ has the highest ratio of teams in certain scenarios, it has the lowest frequency (21.88%) compared to 4H (40.63%). We also ran the same data set as in Table 7. 6 , regardless of the meal break allocation, and the numerical results are shown in Table 7. 7. Again, 6H’ shows a very contradictory result in this sensitivity result. It generates the lowest team ratio 0.78 -
0.87 in the tight trip travelling scenario; meanwhile, the highest team ratio is the 1.
- 1.35 loose trip travelling scenario with small sample sizes. Both these lowest and highest ratios are found by applying the TSH. Table 7. 7 : Working shifts without meal break analysis (4-hour shift ratio base) Tight trip travelling Loose trip travelling IA TSH IA TSH AP- 100 & 250
4H < 6H’ < 8H’ 6H’ < 4H < 8H’ 4H < 6H’ < 8H’ 4H < 8H’ < 6H’
AP-
4H < 6H’ < 8H’ 6H’ < 4H < 8H’
4H < 6H’ < 8H’ 6H’ < 4H < 8H’
AN-
4H < 6H’ < 8H’ 6H’ < 8H’ < 4H
4H < 6H’ < 8H’ 4H < 8H’ < 6H’
AN-
4H < 6H’ < 8H’ 6H’ < 8H’ < 4H 6H’ < 4H < 8H’ 6H’ < 8H’ < 4H
BP-
4H < 6H’ < 8H’ 6H’ < 8H’ < 4H
4H < 6H’ < 8H’ 4H < 8H’ < 6H’
BP-
4H < 8H’ < 6H’ 6H’ < 8H’ < 4H
4H < 6H’ < 8H’ 4H < 8H’ < 6H’
BN-
4H < 8H’ < 6H’ 6H’ < 8H’ < 4H
4H < 6H’ < 8H’ 4H < 8H’ < 6H’
BN-
4H < 8H’ < 6H’ 6H’ < 8H’ < 4H
4H < 6H’ < 8H’ 4H < 8H’ < 6H’
In 21 out of 32 settings the least loading teams are obtained by applying 4H, followed by 6H’ ( 11 out of 32 settings). It is worth to mention that, 6H’ provides the best solution in all tight travelling settings, regardless the demand sizes. On the other hand, 8H’ requires the most teams in half of the settings. The findings presented in Tables 7. 6 and 7. 7 demonstrate that a four-hour shift requires significantly less manpower than both longer working shifts in 32 out of 64 settings. This result makes sense as shorter working shifts have greater flexibility in accommodating complicated multi-trip travelling requirements. The company could consider implementing a four-hour working shift, as it performs much better than the
The summary of the comparative results as a function of the sample size is presented in Table 7. 9. The second row shows the difference of the number of teams required based on the mean value from each set of test problems. The third and fourth rows report the minimum and maximum gap in the differences. In the following two rows, the scenarios where the smallest and biggest differences found are presented. The reported criteria are: the scenario, the heuristic methodology applied, the working shift type, the comparison meal break duration types and the difference interval. The meal break types where the overall smallest and biggest differences in terms of number of loading teams are in seventh and eighth rows. The details are: working shift type, the comparison meal break duration types and the differences interval. There are some negative values of differences found in the numerical results, which means that the number of teams required is less with the longer meal break duration. The last row shows the scenarios, heuristics and data types where the negative values of difference occur. From the table, it can be seen that the smallest difference in terms of the number of teams required is between the 30-minute and the 15-minute meal break in the 8-hour shift, while the largest is in the four-hour shift, without a meal break and with a 15-minute meal break. These results occur in all types of sample sizes. The numerical results also show that there is a huge increase of manpower required from a no meal break allocation to any type of meal break duration, especially when the IA is applied. It is interesting to note the performance analysis performed on the different meal break durations in the six-hour shift. The difference between the 30-minute and the 15-minute meal break in the six-hour shift is the second smallest in comparison. However, the gap in difference greatly increases in BP-750 and BN-750 in the tight trip travelling scenario with the TSH. With all these findings, the company could more suitably choose the meal break type according to sample sizes, scenarios and heuristics. For example, no meal break allocation would be highly recommended in the four-hour working shift due to the possibility of a required increase in manpower. Furthermore, the four-hour working shift is short and mostly applies to casual workers.
Table 7. 9 : Computational results of meal-break duration sensitivity 100 aircraft 250 aircraft 5 00 aircraft 750 aircraft Average interval of differences
[-0.5, 2.63] [-2.78, 4.75] [0.18, 8.08] [-0.33, 21.03]
Min. gap 0 0 0.3 0. Max. gap 4 10.1 18.6 39. Smallest difference settings Loose trip travelling, TSH, four-hour shift: 0 - 1 unit meal break, [-0.2, 0.8] Loose trip travelling, TSH, four-hour shift: 0 - 1 unit meal break, [0.1, 0.6] Loose trip travelling, TSH, four-hour shift: 0 - 1 unit meal break, [-0.6, 1.1] Loose trip travelling, TSH, eight-hour shift: 1 - 2 units meal break, [3.2, 4.5] Biggest difference settings Loose trip travelling, IA, four-hour shift: 0 - 1 unit meal break, [3.1, 4] Tight trip travelling, TSH, eight-hour shift: 0 - 1 unit meal break, [7.9, 10.1] Tight trip travelling, TSH, eight-hour shift: 0 - 1 unit meal break, [16.7, 18.6] Tight trip travelling, TSH, four-hour shift: 0 - 1 unit meal break, [10.9, 39.1] Smallest difference duration types Eight-hour shift: 1 - 2 units meal break, [-0.1, 0.8] Eight-hour shift: 1 - 2 units meal break, [1.55, 1.78] Eight-hour shift: 1 - 2 units meal break, [2.8, 3.25] Eight-hour shift: 1 - 2 units meal break, [3.65, 5.3] Biggest difference duration types Four-hour shift: 0 - 1 unit meal break, [0.83, 2.63] Four-hour shift: 0 - 1 unit meal break, [1.98, 4.75] Four-hour shift: 0 - 1 unit meal break, [3.38, 8.08] Four-hour shift: 0 - 1 unit meal break, [3.95, 21.03] Negative differences
BN & BP, IA,
Six- & eight-hour shifts: 0 - 1unit meal break
BN & BP, IA,
Four-, six- & eight-hour shifts: 0 - 1 unit meal break
BN & BP, IA,
Four-, six- & eight-hour shifts: 0 - 1 unit meal break
BN & BP, IA,
Four-, six- & eight-hour shifts: 0 - 1 unit meal break
