Optimal opportunistic tamping scheduling for railway track geometry

Figures & data

Figure 1. Example of a UH2 defect of the longitudinal level.

Table 1. Maintenance limits defined in the European standard EN 13848–5 (2008) and in Trafikverket (2015).

EN 13848-5 limits	Trafikverket’s limits
Alert limit	Planning limit
Intervention limit	UH1 limit (lower bound for corrective maintenance)
	UH2 limit (upper bound for corrective maintenance)
Immediate action limit	Critical limit

Figure 2. (a) Preliminary tamping schedule without grouping, (b) group maintenance activities achieved by tamping postponement and (c) group maintenance activities achieved by earlier tamping.

Figure 3. Schematic description of maintenance limits and maintenance zones.

Figure 4. The procedure for achieving an optimal tamping schedule.

Figure 5. The evolution of the SDLL in multiple maintenance cycles.

Figure 6. Histograms of the length of the track, the initial degradation value and the degradation rate.

Table 2. Cost parameters.

Description	Parameter	Cost (SEK)
Fixed cost of maintenance (per maintenance window)	$f$	150,000
Cost of preventive tamping (per metre)	$C_{pm}$	25
Cost of the probability of isolated defects	$C_{\mathit{iso}}$	11,000
Cost of unused life due to earlier performance of tamping action	$C_{UL}$	3,000

Table 3. Initial parameters to develop the model.

Description	Parameters	Value
Time horizon	T	3 years
Maintenance windows	t	6 months
Possession time for tamping work	$T^P$	6 hours
Warm-up and cool-down time	$T_{cw}$	20 mins.
Speed of tamping machine during tamping	$V_{\mathit{tamp}}$	1 km/h
Speed of tamping machine during travelling	$V_{tr}$	80 km/h
Lower bound of preventive limit	plan	1.2 mm

Table 4. Parameters of the genetic algorithm.

Parameter	Value
Size of the population	150
Number of generations	300
Cross-over percentage	0.9
Mutation percentage	0.5
Mutation probability	0.005

Table 5. Detailed results for the optimal tamping schedule.

Maintenance window	(1(1) $$C_{X,Z}=\mathit{minimize}\mathrm{ }{C}_{X,Z}^{\mathit{Tamping}}+C_{X,Z}^{UH2\mathrm{ }\mathit{defects}}+C_{X,Z}^{\mathit{Unused}\mathrm{ }\mathit{life}}$$(1) )	(2(2) $${\mathit{SDLL}}_{s,n}\left(t\right)={\mathit{SDLL}}_{s,n}^0.e^{b_s^0{(1+\omega )}^{(n-1)}\times (t-{T_{lt}}_s)}$$(2) )	(3(3) $$R_{s,n}=\frac{{\mathit{SDLL}}_{s,n}^0}{{\mathit{SDLL}}_{s,n}\left(t_{\mathit{tamp}}^{-}\right)}$$(3) )	(4(4) $${\widehat{R}}_{s,n}=\{\begin{array}{c}\alpha +\beta {\mathit{SDLL}}_{s,n}\left(t_{\mathit{tamp}}^{-}\right), n=1\\ R_{s,1}.{(1-\phi )}^{n-1}, n>1\end{array}$$(4) )	(5(5) $$P\left(Y_{s,t}=1\left|{\mathit{SDLL}}_{s,n}\right(t)\right)=\pi \left({\mathit{SDLL}}_{s,n}\right(t\left)\right)=\frac{e^{{\beta }_0+{\beta }_1{\mathit{SDLL}}_{s,n}\left(t\right)}}{1+e^{{\beta }_0+{\beta }_1{\mathit{SDLL}}_{s,n}\left(t\right)}}$$(5) )	(6)	Total
First scenario (base case)
# tamping	14	0	10	0	18	0	42
Total length tamped (metres)	2354	0	1657	0	3357	0	7368
Fixed cost							4.500˟10^5
Preventive tamping cost							1.473˟10^5
Corrective tamping cost							1.671˟10^5
Unused life cost							2.601˟10^5
Total cost (SEK)							10.246˟10^5
Second scenario
# tamping	9	0	7	0	0	0	16
Total length tamped (metres)	1432	0	1265	0	0	0	2697
Fixed cost							3.000˟10^5
Preventive tamping cost							0.539˟10^5
Corrective tamping cost							2.210˟10^5
Unused life cost							1.188˟10^5
Total cost (SEK)							6.251˟10^5
Third scenario
# tamping	9	0	0	4	0	0	13
Total length tamped (metres)	1457	0	0	711	0	0	2168
Fixed cost							3.000˟10^5
Preventive tamping cost							0.434˟10^5
Corrective tamping cost							2.540˟10^5
Unused life cost							1.254˟10^5
Total cost (SEK)							7.228˟10^5

Figure 7. Mean probability of occurrence of isolated defects for the whole line for different For Peer Review Only scenarios.

Figure 8. The optimal tamping schedule for the base case scenario.

Table 6. The results of optimal scheduling for the base case scenario, with and without considering the change in the degradation rate parameter.

Change in degradation rate after tamping	# tamping	Length of tamped track sections (metres)	Total cost
$\omega =0$	40	7026	$10.010\mathrm{˟}{10}^5$
$\omega =5\mathrm{\%}$	42	7368	$10.246\mathrm{˟}{10}^5$

Figure 9. Optimal tamping schedules for the base case scenario for a fixed cost of maintenance equal to a) 10,000 SEK, b) 50,000 SEK and c) 150,000 SEK.

Table 7. The results of the model with and without the unused life included in the objective function.

Maintenance window	(1(1) $$C_{X,Z}=\mathit{minimize}\mathrm{ }{C}_{X,Z}^{\mathit{Tamping}}+C_{X,Z}^{UH2\mathrm{ }\mathit{defects}}+C_{X,Z}^{\mathit{Unused}\mathrm{ }\mathit{life}}$$(1) )	(2(2) $${\mathit{SDLL}}_{s,n}\left(t\right)={\mathit{SDLL}}_{s,n}^0.e^{b_s^0{(1+\omega )}^{(n-1)}\times (t-{T_{lt}}_s)}$$(2) )	(3(3) $$R_{s,n}=\frac{{\mathit{SDLL}}_{s,n}^0}{{\mathit{SDLL}}_{s,n}\left(t_{\mathit{tamp}}^{-}\right)}$$(3) )	(4(4) $${\widehat{R}}_{s,n}=\{\begin{array}{c}\alpha +\beta {\mathit{SDLL}}_{s,n}\left(t_{\mathit{tamp}}^{-}\right), n=1\\ R_{s,1}.{(1-\phi )}^{n-1}, n>1\end{array}$$(4) )	(5(5) $$P\left(Y_{s,t}=1\left|{\mathit{SDLL}}_{s,n}\right(t)\right)=\pi \left({\mathit{SDLL}}_{s,n}\right(t\left)\right)=\frac{e^{{\beta }_0+{\beta }_1{\mathit{SDLL}}_{s,n}\left(t\right)}}{1+e^{{\beta }_0+{\beta }_1{\mathit{SDLL}}_{s,n}\left(t\right)}}$$(5) )	(6(6) $$\min E\left(C_{X,Z}\right)=\sum _{t\in T}\left(z_t.f+\sum _{s\in S}{x}_{s,t}.L_s.C_{pm}\right)+{\sum }_{t\in T}{\sum }_{s\in S}{P}_{s,t}.C_{\mathit{iso}}+{\sum }_{t\in T}{\sum }_{s\in S}{UL}_{s,t}.x_{s,t}.C_{UL}$$(6) )	Total
Model without unused life
# tamping	17	14	0	0	13	0	43
Total length tamped (metres)	2796	2578	0	0	2274	0	7468
Consumed possession time	6.0	5.4	0	0	4.4	0	15.8
Model with unused life (base case)
# tamping	14	0	10	0	18	0	42
Total length tamped (metres)	2354	0	1657	0	3357	0	7368
Consumed possession time	5.6	0	4.5	0	5.8	0	15.9

Figure 10. Effect of the tamping speed and travelling speed of the tamping machine on the total working performance.

Table 8. Effect of the tamping speed of the tamping machine on the total maintenance costs.

Tamping speed (km/h)	Total maintenance costs (SEK)
0.5	13.101˟10^5
1.0 (base case)	10.246˟10^5
1.5	10.246˟10^5

EN 13848-5. (2008). Railway applications – track – track geometry quality. Part 5: Geometric quality levels. Brussels: European Committee for Standardization (CEN).

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Trafikverket. (2015). Banöverbyggnad – spårläge – krav vid byggande och underhåll (Track superstructure – track geometry – requirements for renewal and maintenance [in Swedish]). Report No. TDOK 2013:0347 v3.0. Borlänge, Sweden: Trafikverket.

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