Optimized phased planning for dynamic rehabilitation of integrated municipal infrastructure

Scale factor

Shape parameter

Time span

γ

Location parameter

Rehabilitation program

Area of corridor

Percentage full coordination for the scenario of for a, b, and c systems

Percentage partial coordination for the scenario of

Cost of leakage detection

Replacement cost of pipes

Repair cost

Resurfacing cost

Surface overlay cost

Cost of corridor i for the system s in phase

Diameter

Health reliability threshold of element i of the system s

Health reliability of pipe i at the time j

Health reliability of pipe i at the pristine time

Health reliability of pipe i at the time j

Leakage emitter value

Length of corridor

Number of corridors

Number of phases

Number of systems

Phase indicator

Type of intervention

Discount rate

System indicator

Planning horizon year

Unit cost for pipe replacement

Unit cost for road resurfacing

Unit cost for road surface overlay

Number of weeks

Intervention decision in corridor i for the system a in phase p

Rehabilitation decision in corridor i for the system b in phase p

Intervention decision in corridor i for the system c in phase

The well-being and economic growth of societies worldwide are significantly influenced by urban infrastructure systems. As an example, an investment of $1 billion in infrastructure not only creates 16,700 jobs but also enhances the gross domestic product (GDP) by $1.6 billion, as reported by Finance Canada. However, these systems are continuously exposed to degradation due to aging, excessive use, and natural hazards (De Iuliis et al. 2019; Liu & Song 2020), leading to deteriorated infrastructures in numerous municipalities globally. A notable example is the state of Canada's municipal infrastructures, where one-third is categorized as being in fair, poor, or failing condition (Abu-Samra et al. 2018). Consequently, comprehensive renewal plans, encompassing repair and rehabilitation interventions, become imperative to effectively and economically preserve the value and performance of infrastructure assets.

Both qualitative and quantitative approaches for solving engineering and environmental problems can be found in the literature. Among qualitative approaches, observation methods, surveys, questionnaires, and various planning documents are essential for solving a problem (Sukri et al. 2023). Quantitative approaches use complex mathematical models including differential equations, artificial intelligence, and optimization algorithms (Feng et al. 2022; Bachtiar et al. 2023; Khalaf et al. 2023; Moayedi et al. 2024). The focus of the current study is on using a quantitative approach for solving complex intervention planning of urban infrastructure systems.

Over the past two decades, attention has significantly increased toward system thinking approaches for addressing asset management challenges, including rehabilitation, maintenance, and operation planning (Kerwin & Adey 2021; Okwori et al. 2021; Daulat et al. 2022). These approaches involve the strategic coordination and integration of infrastructural activities across various interdependent systems to achieve cost savings, sustainability, and reliability. Capital renewal plans pose a complex, dynamic, and multi-criteria decision-making challenge for asset managers. Researchers have frequently turned to optimization algorithms as preferred tools for addressing asset-related issues (Amador & Magnuson 2011; Rashedi & Hegazy 2015; Rashedi & Hegazy 2016; Van Dijk & Hendrix 2016; Alinizzi et al. 2018). The dynamic nature of these problems, where infrastructure properties such as physical states and design factors (like customer demands) evolve over time, calls for the use of dynamic optimization approaches, such as dynamic programming. Consequently, this work developed a phased design and construction approach that employs optimization models, offering a more adaptive solution compared with static approaches.

As indicated in Table 1, researchers have used the phased design and construction approach in addressing both multi-utility and single-utility asset management problems. Nevertheless, it is noteworthy that the phase duration has consistently been treated as a fixed parameter in optimization problems and typically ranges from 1 to 25 years.

Looking at the different number of phases and their durations found in the literature, the question of whether the number of phases (and consequently their duration) is suitable to be treated as a decision variable to produce optimal master plans arises, and furthermore, what could be their role in the rehabilitation strategies for the renewal plans? Is it better to upgrade the networks in many short phases by implementing minor rehabilitation actions (e.g., leakage repair for pipes and surface overlay of pavements) or a few longer phases by major rehabilitation actions (pipe replacement or resurfacing of pavements)? This study aims to explore this research gap in the literature by solving a real-world rehabilitation problem in Montreal City, Canada, for three interdependent urban systems, which are the water, sewer, and road networks. The case study was first presented by Abu-Samra et al. (2020). They addressed a rehabilitation problem encompassing water, sewer, and road networks within an asset management framework. Our work uses their case study with a minor change through reconstructing water leakages in pipes. The approach will demonstrate the benefits of phased design optimization when applying minor and major rehabilitation actions on the three interdependent networks. Abu-Samra et al. (2020) work's approach involved a phased planning method, dividing a 25-year planning horizon into 25 one-year phases. The present study formulates a dynamic optimization programming model, treating the phase duration as a decision variable, for the rehabilitation planning of interdependent urban infrastructures. Results from the new methodology are compared with those in the referenced work, resulting from fixed phase planning, to understand the role of phase duration in achieving optimal results.

The primary contribution of this paper is the introduction of variable phase numbers within the framework of phased intervention planning for three urban infrastructure systems. This dynamic approach addresses a significant gap in the literature concerning the integrated management of interdependent infrastructure systems. Our method represents a novel advancement in phased planning optimization, enhancing the phased design and construction strategy for interdependent systems. Additionally, the genetic algorithm employed has been specifically developed to manage three-dimensional solutions of varying sizes within the mating pool.

Equation (1) represents the Weibull functional form where represents the health reliability of pipe i, where is the number of corridors (every corridor represents a segment in the network including a WN pipe beside an SN pipe on which there is an RN pavement) at the times of ⁠, (shown in Figure 1); stands for the health reliability of pipe i at the pristine time (usually considered 100%); is the age of pipe at the time ⁠; α and β are the scale factors (years, defined based on the service life span of pipes depending on the type of materials which could be PVC, cast iron, cement, etc.) and shape parameter (>0); the third parameter of Weibull probability distribution, location parameter (γ) is equal to 0 in this study.

Equation (2) represents a nonlinear deterioration mathematical model where stands for the health reliability of road i at the time ⁠; is the age of road i at the time (the reference is year 0); and is the health reliability of road i at the year 0.

Based on the health condition of every element at a time, different decisions for interventions can be made. The current health condition of the element could be satisfactory when its deterioration over a target phase does not cause a complete failure (health reliability < threshold); hence, a decision for no intervention could be made; otherwise, rehabilitation activities should be pursued. In the current study, the interventions on the elements could be minor or major actions shown in Table 2. Every action has its impact value for upgrading the physical status of pipes and road pavements. Health improvements after minor actions depend on the current condition of the elements. For example, repairing cracks in water pipes with current reliability of 90% would not give the same improvement if the current reliability is 10%. However, for the sake of simplicity, it is assumed to be 10% for minor interventions regardless of the initial reliability and 100% for major interventions (Abu-Samra et al. 2020).

The replacement cost of a pipe will depend on the diameter of the pipe with as the replacement cost in €/m (note: the prices with € in this study are changed to CAN$ with the exchange rate of 1.44 Canada dollar/€), and is the length of corridor in m. As regards the leak, the k value grows with time based on Equation (9), where is the initial value of the k coefficient and w is the number of weeks since the start of the work. It is assumed that after pipe replacement, the pipe does not leak until the last phase, and therefore, the k value becomes 0; in the repair case, the k value neither gets to 0 nor grows and its value becomes constant over the target phase.

Regarding the road costs, and represent the resurfacing and surface overlay costs for road pavements, respectively, where and are unit costs for the two mentioned activities in the area of interest (CAN$/m²), and A is the area of the corridor (m²).

As regards the constraint of the optimization, except for meeting the health threshold in every phase of the optimization, there is a practical consideration such that in every corridor, once a major action has been taken on an element, the element does not get renovated by the major action again (for example, multiple pipe replacements is not possible over the phases).

Abu-Samra et al. (2020) proposed three approaches for coordinating the infrastructural activities between the three systems of water, sewer, and road. The approach assumes that in every corridor, the intervention activities of different systems could be coordinated to minimize the asset management costs by synchronizing activity for different systems (Table 3).

To demonstrate the functionality of the methodology, the proposed optimization approach is applied to a 9 km stretch from the city of Montreal's roads, water, and sewer networks (Abu-Samra et al. 2020). The network comprises 20 corridors, each of which has pipes and roads with different properties shown in Table 4. The total number of decisions for intervention actions in ‘no coordination’, ‘partial coordination’ and ‘full coordination’ scenarios in a corridor are 9, 27, and 27, respectively, while there are 25 decisions on the number of phases within the planning horizon period. Therefore, for 1, 2, …, and 25 phase scenarios, the number of solutions are equal to (9 + 27 + 27)²⁰, 2 × (9 + 27 + 27)²⁰, and 25 × (9 + 27 + 27)²⁰, respectively; hence, the search space size is equal to 325 × (9 + 27 + 27)²⁰.

Table 4

Sample from the central database for Montreal's infrastructure systems (Abu-Samra et al. 2020)

General .		Road network .				Water network .			Sewer network .
Corridor ID no. .	Corridor length (m) .	Number of lanes .	Lane width .	Section area (m2) .	Current condition (%) .	Year of installation .	Pipe diameter (mm) .	(×10−4) .	Year of installation .	Pipe diameter (mm) .	(×10−4) .
1	370	3	3	3,330	90	1953	450	8.00	1920	600	3.60
2	370	4	3	4,440	70	1982	150	0.15	1900	525	4.10
3	452	4	3	5,424	85	1976	250	8.10	1893	375	27.70
4	393	2	3	2,385	65	1958	200	14.00	1950	300	0.67
5	419	3	3	3,771	70	1965	150	2.70	1960	250	7.50
6	766	4	3	9,192	90	1991	100	2.90	1970	150	7.90
7	451	4	3	5,412	70	1992	500	427.10	1980	200	7.70
8	311	2	3	1,866	85	1977	500	2.90	1990	450	6.50
9	425	4	3	5,100	65	1982	350	2.20	2000	200	0.73
10	783	4	3	9,396	70	1991	500	0.10	1975	525	0.39
11	318	3	3	2,862	90	1972	500	0.09	1943	375	0.68
12	162	4	3	1,944	70	1960	250	8.70	1955	300	3.40
13	498	4	3	5,976	85	1979	250	5.00	1965	250	8.70
14	686	4	3	8,232	65	1953	100	5.20	1975	150	8.40
15	207	2	3	1,242	70	1960	450	3.30	1985	200	0.91
16	715	3	3	6,435	90	1977	200	3.10	1905	450	10.80
17	270	2	3	1,620	70	1986	150	14.90	1965	200	10.40
18	217	2	3	1,302	85	1992	350	8.70	1968	600	15.10
19	519	2	3	3,114	65	1975	300	4.00	1978	525	18.80
20	560	4	3	6,720	70	1987	150	0.85	1982	600	6.90

General .		Road network .				Water network .			Sewer network .
Corridor ID no. .	Corridor length (m) .	Number of lanes .	Lane width .	Section area (m2) .	Current condition (%) .	Year of installation .	Pipe diameter (mm) .	(×10−4) .	Year of installation .	Pipe diameter (mm) .	(×10−4) .
1	370	3	3	3,330	90	1953	450	8.00	1920	600	3.60
2	370	4	3	4,440	70	1982	150	0.15	1900	525	4.10
3	452	4	3	5,424	85	1976	250	8.10	1893	375	27.70
4	393	2	3	2,385	65	1958	200	14.00	1950	300	0.67
5	419	3	3	3,771	70	1965	150	2.70	1960	250	7.50
6	766	4	3	9,192	90	1991	100	2.90	1970	150	7.90
7	451	4	3	5,412	70	1992	500	427.10	1980	200	7.70
8	311	2	3	1,866	85	1977	500	2.90	1990	450	6.50
9	425	4	3	5,100	65	1982	350	2.20	2000	200	0.73
10	783	4	3	9,396	70	1991	500	0.10	1975	525	0.39
11	318	3	3	2,862	90	1972	500	0.09	1943	375	0.68
12	162	4	3	1,944	70	1960	250	8.70	1955	300	3.40
13	498	4	3	5,976	85	1979	250	5.00	1965	250	8.70
14	686	4	3	8,232	65	1953	100	5.20	1975	150	8.40
15	207	2	3	1,242	70	1960	450	3.30	1985	200	0.91
16	715	3	3	6,435	90	1977	200	3.10	1905	450	10.80
17	270	2	3	1,620	70	1986	150	14.90	1965	200	10.40
18	217	2	3	1,302	85	1992	350	8.70	1968	600	15.10
19	519	2	3	3,114	65	1975	300	4.00	1978	525	18.80
20	560	4	3	6,720	70	1987	150	0.85	1982	600	6.90

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In the database, there are some missing values for the parameter k, which were reconstructed stochastically and validated for every pipe based on literature values (Mottahedin et al. 2023).

The rehabilitation is planned for a planning horizon of 25 years (⁠ = 25) and the threshold values are considered 50 and 65% for the pipes and pavements for the whole period, respectively (Abu-Samra et al. 2020). At the year 0, the health reliability values for WN, SN, and RN are 83, 69, and 76%, respectively. The condition health of the sewer is considered the most critical because the sewer pipes in corridors 1, 2, 3, and 16 have a health reliability measure below the threshold. The pipe material is polyethylene with a service life of 100 years and the pavements including asphalts, basements, and subbasements have a 25-year life span.

Figure 6(a) displays the optimization outcomes, with the optimal phasing favoring a design of four phases, each lasting 6.25 years (or 6 years and 3 months). This configuration results in an integrated rehabilitation plan totaling approximately $21.03 M in net present value cost. It is evident that the scenario with incurs the minimum cost; nonetheless, the corresponding solution proves infeasible since initiating the rehabilitation plan in year 0 fails to maintain the health condition of elements above the relevant thresholds for the duration of 25 years.

Figure 6(b) illustrates how the number of phases impacts the coordination of rehabilitation activities across different systems. Generally, corridors have a higher chance of partial coordination compared with full coordination. Moreover, fewer phases increase the likelihood of coordinating activities in corridors compared with scenarios with a higher number of phases. For instance, planning with just one phase results in numerous corridors undergoing rehabilitation actions in year 0, as intervention decisions are based on the health status of systems at year 25. However, aside from the planning horizon, this issue also depends on the initial health conditions of elements from different systems within each corridor. For instance, if a corridor contains a sewer pipe with 100% health reliability but poor conditions in its water pipe and street section, full coordination might become unlikely, while partial coordination is more probable.

As evident, the coordination potential of scenario stands at approximately 18% (number of corridors with coordinated activities/maximum possible number of coordinations in all corridors). Consequently, while scenario proves optimal in terms of minimizing the cost of an integrated rehabilitation plan, it falls short in terms of coordinating activities. This limitation arises from the formulation of the cost model in this study, which does not incorporate social and environmental costs. While this is a significant assumption and simplification, the consideration of the number of phases as a decision variable in the optimization process (a novelty in the current study) remains crucial for attaining an optimal solution.

The detailed scheduling of the rehabilitation plans for the optimal solution, and the reference work's number of phases are shown in Table 6.

Table 6

Optimum infrastructures’ rehabilitation planning and comparison with the reference one

Year .	(Abu-Samra et al. 2020) .						(optimum, current study) .
	Water .		Sewer .		Road .		Water .		Sewer .		Road .
	Maj (M) .	Min (K) .	Maj (M) .	Min (K) .	Maj (M) .	Min (K) .	Maj (M) .	Min (K) .	Maj (M) .	Min (K) .	Maj (M) .	Min (K) .
0	0	0	7.70	0	0	616.21	0	0	7.71	0	4.55	0
1	0	0	0	0	0	1,109.83	N/A	N/A	N/A	N/A	N/A	N/A
2	0	0	0	0	0	592.28	N/A	N/A	N/A	N/A	N/A	N/A
3	0	0	0	0	0	0.00	N/A	N/A	N/A	N/A	N/A	N/A
4	0	0	0	0	0	1,615.06	N/A	N/A	N/A	N/A	N/A	N/A
5	0	0	0	0	0	0	N/A	N/A	N/A	N/A	N/A	N/A
6	0	0	0	4.93	0	1,005.18	0	0	0.69	0	1.10	631.77
7	0	0	0	0	0	952.04	N/A	N/A	N/A	N/A	N/A	N/A
8	0	0	0	0	0	0	N/A	N/A	N/A	N/A	N/A	N/A
9	0	0	0	0	0	2,061.09	N/A	N/A	N/A	N/A	N/A	N/A
10	0	0	0	0	0	391.63	N/A	N/A	N/A	N/A	N/A	N/A
11	0	0	0	0	0	1,485.47	N/A	N/A	N/A	N/A	N/A	N/A
12	0	0	0	4.40	0	861.82	0	8.81	0.48	3.82	5	0
13	0	0	0	3.74	0	0	N/A	N/A	N/A	N/A	N/A	N/A
14	0	2.75	0	0	0	1,761.30	N/A	N/A	N/A	N/A	N/A	N/A
15	0	0	0	0	0	457.85	N/A	N/A	N/A	N/A	N/A	N/A
16	0	0	0	0	0	1,345.44	N/A	N/A	N/A	N/A	N/A	N/A
17	0	4.79	0	0	0	780.58	N/A	N/A	N/A	N/A	N/A	N/A
18	0	0	0	7.34	0	0	N/A	10.32	N/A	3.03	0.86	0
19	0	0	0	3.33	0	1,690.82	N/A	N/A	N/A	N/A	N/A	N/A
20	0	2.61	0	0	0	3,212.70	N/A	N/A	N/A	N/A	N/A	N/A
21	0	2.37	0	0	0	471.75	N/A	N/A	N/A	N/A	N/A	N/A
22	0	2.81	0	2.82	0	1,439.60	N/A	N/A	N/A	N/A	N/A	N/A
23	0	4.24	0	0	0	0	N/A	N/A	N/A	N/A	N/A	N/A
24	0	4.17	0	6.53	0	1,445.13	0	0	0	0	0	0
Sum.	0	23.75	7.70	33.10	0	20,405.85	0	18.40	8.88	6.85	11.50	631.77
TNPVC	$28.16M (CAD)						$21.03M (CAD)

Year .	(Abu-Samra et al. 2020) .						(optimum, current study) .
	Water .		Sewer .		Road .		Water .		Sewer .		Road .
	Maj (M) .	Min (K) .	Maj (M) .	Min (K) .	Maj (M) .	Min (K) .	Maj (M) .	Min (K) .	Maj (M) .	Min (K) .	Maj (M) .	Min (K) .
0	0	0	7.70	0	0	616.21	0	0	7.71	0	4.55	0
1	0	0	0	0	0	1,109.83	N/A	N/A	N/A	N/A	N/A	N/A
2	0	0	0	0	0	592.28	N/A	N/A	N/A	N/A	N/A	N/A
3	0	0	0	0	0	0.00	N/A	N/A	N/A	N/A	N/A	N/A
4	0	0	0	0	0	1,615.06	N/A	N/A	N/A	N/A	N/A	N/A
5	0	0	0	0	0	0	N/A	N/A	N/A	N/A	N/A	N/A
6	0	0	0	4.93	0	1,005.18	0	0	0.69	0	1.10	631.77
7	0	0	0	0	0	952.04	N/A	N/A	N/A	N/A	N/A	N/A
8	0	0	0	0	0	0	N/A	N/A	N/A	N/A	N/A	N/A
9	0	0	0	0	0	2,061.09	N/A	N/A	N/A	N/A	N/A	N/A
10	0	0	0	0	0	391.63	N/A	N/A	N/A	N/A	N/A	N/A
11	0	0	0	0	0	1,485.47	N/A	N/A	N/A	N/A	N/A	N/A
12	0	0	0	4.40	0	861.82	0	8.81	0.48	3.82	5	0
13	0	0	0	3.74	0	0	N/A	N/A	N/A	N/A	N/A	N/A
14	0	2.75	0	0	0	1,761.30	N/A	N/A	N/A	N/A	N/A	N/A
15	0	0	0	0	0	457.85	N/A	N/A	N/A	N/A	N/A	N/A
16	0	0	0	0	0	1,345.44	N/A	N/A	N/A	N/A	N/A	N/A
17	0	4.79	0	0	0	780.58	N/A	N/A	N/A	N/A	N/A	N/A
18	0	0	0	7.34	0	0	N/A	10.32	N/A	3.03	0.86	0
19	0	0	0	3.33	0	1,690.82	N/A	N/A	N/A	N/A	N/A	N/A
20	0	2.61	0	0	0	3,212.70	N/A	N/A	N/A	N/A	N/A	N/A
21	0	2.37	0	0	0	471.75	N/A	N/A	N/A	N/A	N/A	N/A
22	0	2.81	0	2.82	0	1,439.60	N/A	N/A	N/A	N/A	N/A	N/A
23	0	4.24	0	0	0	0	N/A	N/A	N/A	N/A	N/A	N/A
24	0	4.17	0	6.53	0	1,445.13	0	0	0	0	0	0
Sum.	0	23.75	7.70	33.10	0	20,405.85	0	18.40	8.88	6.85	11.50	631.77
TNPVC	$28.16M (CAD)						$21.03M (CAD)

Maj, major; Min, minor; Sum.: this is the summation of net present value costs of every year cost; TNPVC, total net present value cost (the summation of total costs of water, sewer, and road rehabilitation project); N/A, not available.

The minor and major scale of rehabilitation costs are thousands (K) and millions (M).

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Comparing the result obtained in the current study (when the number of phases is among the optimization decision variables) with the reference case (with a fixed number of 25 phases, each with a one-year interval) shows that the number of phases plays a critical role in cost saving of the whole rehabilitation project such that around $7 M saving is achieved (about 25% saving). Planning for multiple short-term phases and investing more in minor rehabilitation actions is not a good approach at least for the current case study. In other words, larger, fewer, and more timely targeted investments are better than more frequent and smaller investments, particularly in the case of road rehabilitation. It is worth mentioning that the original study also used optimization, which means that more than 25% can be achieved in ordinary master planning where optimization may not have been used. Another benefit of a decrease in the number of interventions (35 against 12 for the scenarios of and ⁠, respectively, about 65% fewer interventions) are savings in the public nuisance (including traffic, water, and sewer services interruptions), which translates to indirect costs not evaluated in this study.

Considering the results, the costs related to road activities exceed those related to water and sewer activities (millions vs thousands of dollars), showing the important role of optimal road intervention planning in minimizing the total integrated rehabilitation cost. This could help asset managers prioritize asset management activities.

Comparing Figure 7(a) and 7(c), in both reference and optimal plans, similarly larger investments are required for the sewer system than the water system indicating the more critical condition of the sewer system than the water system (the major rehabilitation cost is truncated in Figure 7(a) and 7(c) for the clear comparability demonstration with minor rehabilitation cost). This is due to the critical condition of sewer pipes in the year 0 in some corridors (as seen in Table 4, some sewer pipes have more than 100 years and urgent action is needed for their reclamation). This shows that the initial condition of systems is an important factor in optimal decision-making for the integrated rehabilitation of systems where the impacts of phase durations and other variables are minor. For example, in both reference and optimal plans, due to the fact the age of the water system is lower than the sewer system, multiple minor rehabilitation actions have been scheduled in the water system through some phases (without any major rehabilitation actions). The total present value costs are $23.75 K and $18.40 K (CAD) in the reference and optimal plan, respectively, achieving around 22% cost savings. The case study is done on a small part (only 9 km) of the city of Montreal network, which makes $23.75 K and $18.40 K very small investments compared with what the city is spending on repairing leaks in the water networks.

In the reference plan, the sewer system has undergone a major rehabilitation action with around $7.70 M in the first phase. Its health condition is then maintained above the threshold by minor rehabilitation activities over several phases with the total present value costs of $33.10 K. The corresponding costs for the optimal plan are $8.88 M and $6.85 K, respectively, meaning around 15% excessive cost on the total rehabilitation investment on the sewer system.

Figure 7(b) and 7(d) compares the road rehabilitation scheduling over phases in the optimal and reference plan. In the optimal plan, major and minor rehabilitation actions have been planned with a total net present value cost of $11.50 M and $631.77 K, respectively. In the reference plan, frequent minor rehabilitation interventions have been designed through many short phases. That makes the total cost around $20.40 M, meaning 41% saving the rehabilitation cost on the road system.

Considering the proposed methodology and the results of the current study, the question may arise of whether fixed intervals bring about an overall optimal solution. For example, while 4 years and 3 months is an optimal duration in phase 1, another duration could be optimal for the second phase. This issue together with the fixed number of phases have been two important challenges and limitations of the phased design and construction approach within the application of the evolutionary optimization for solving many types of engineering problems. This study proposed an approach to overcome the challenge of a fixed number of phases while the challenge of fixed intervals remains an important question and will be investigated in future research work.

In the literature, interval optimization within the dynamic intervention planning of urban infrastructure systems has been missing to date. This study introduced phased design and construction to the rehabilitation of three spatially interdependent urban infrastructure systems: water, sewer, and road. In this approach, the design period is divided into a different number of equally long phases where each scenario includes a particular phase duration. The integrated system is then optimized to find the least cost design for intervention planning while keeping the health condition of the systems above the associated thresholds. Not only the interventions were among the decision variables, but also the duration of phases was determined in the process of optimal decision-making. This has been overlooked in previous studies in the context of multi-phase design and construction approaches to the rehabilitation of infrastructure systems. The case study shows that the timing of the rehabilitation can play a major role in finding the minimum investment cost. In this case, four phases, each lasting about 6 years led to around 25% cost savings for the rehabilitation of Montreal's urban infrastructures compared with the reference work's plan (Abu-Samra et al. 2020) where there were 25 phases, each one year long. In the current case study, the longer duration and larger investment interventions (as opposed to the shorter duration and smaller investments) contribute to an optimal integrated rehabilitation plan. Hence, decision-makers can use this important finding for better planning of their projects to achieve even more cost savings than demonstrated in the current study due to it being applied to a small part of a large interdependent system.

Another interesting finding of the current study is related to the impacts of the number of phases (and their duration) on making synchronized rehabilitation planning among different systems. Shorter phases could be a barrier to achieving a coordinated intervention plan for different systems. This depends significantly on the deterioration rates of different elements in considered systems (pipes and pavements), which were tackled through a deterministic approach. Future work will include an exploration of the impact of variable phase durations, the effects of applying uncertainty evaluation to the deterioration models, and the addition of operating, social and environmental costs to the cost model.

The authors wish to extend their heartfelt appreciation to the individuals, organizations, and funding agencies in Austria who have supported this research. Our deep appreciation goes to the Vice Rectorate for Research at the University of Innsbruck, the Austrian Academy of Sciences (ÖAW), and the Federal Ministry of Education, Science and Research (BMBWF) in the Republic of Austria (OeAD-GmbH – Austria's Agency for Education and Internationalisation; Mobility Programmes and Cooperation) for providing financial support through the Early-Stage Funding, DOC Fellowship (AP845023), and Marietta Blau scholarship which have facilitated the doctoral education of Mr Amin Minaei (the first author) and his research stays at KWR Water Research Institute in the Netherlands and the University of Adelaide in Australia. Moreover, Dragan Savic has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. [951424]).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.