Abstract

Water management and preventing water shortage require accurate planning with attention to the importance of urban water. The problems ahead include the increase in demand and reduction in water supply resources due to factors that cause uncertainties and the high cost of water supply infrastructures. Most studies in urban water management consider only a single criterion. However, in this research, two objective functions, namely cost minimization and per capita water consumption maximization, were used simultaneously. A portfolio approach based on the balance of water supply and demand was developed taking uncertainty into account. Then, the problem was solved using a hybrid robust–stochastic optimization approach. The results showed the selected supply augmentation and demand management options in each stage under dry, normal, and wet year scenarios.

HIGHLIGHTS

  • A bi-objective optimization model is presented to manage water resources.

  • Costs and water consumption per capita are taken into account.

  • The balance of uncertain supply and demand is investigated using a hybrid stochastic–robust optimization approach.

  • Supply augmentation and demand management options are selected.

  • The results are provided for dry, normal, and wet year scenarios.

Graphical Abstract

Graphical Abstract
Graphical Abstract

INTRODUCTION

Water scarcity has gradually turned into a critical issue and hence the need for proper use of water resource management techniques is more emphasized than ever. There is an inherent uncertainty in water supply and demand. Consequently, urban water management and planning should take a comprehensive standpoint on water supply and demand (Morgan Torabi & Dedekorkut-howes 2020). In this regard, Mortazavi-Naeini et al. (2014), in order to expand scheduling capacity, proposed a multi-objective optimization model for a dynamic planning problem of urban water resource systems. Nowadays, the uncertainty has even more intensified with climate change and the security of supply sources has become more crucial. Paton et al. (2014) introduced an integrated framework to assess urban water supply security. They developed a multi-objective optimization model to minimize the cost and to maximize the supply security, simultaneously. Given the importance of water supply and demand with budget limits, Miao et al. (2014) proposed a plan for water resource management systems under uncertainty using a new fuzzy gap. The aim was to evaluate the cost efficiency of resources. In terms of the availability of water resources, Kanta & Zechman (2014) considered the structure of integrated adaptive systems as interactions among the hydrological cycle, politics, and consumers. They formulated urban water supply as a Complex Adaptive System (CAS) and employed random consumer demand alongside an Agent-Based water supply Model (ABM). In the ABM, governments were considered as agents that chose water conservation and transfer strategies. On the other hand, consumers acted as agents influenced by various conservation programs that chose technologies and water conservation behaviors.

Introduced by Markowitz (1952), Modern Portfolio Theory (MPT) was initially used in financial research. The Markovitz model is a classic approach to portfolio optimization based on two incompatible objective functions, namely minimizing the preferred risk level (displayed by variance) and maximizing the expected return. Beuhler (2006) employed portfolio theory in water resources management. In this study, the integrated resource planning for regional water supply focused on water requirements by combining water resources. The combination of the optimal resource in the past was determined by selecting the least costly resource without taking into account the risk. However, managers recognized the value of risk reduction using a combination of various resources.

While numerous researchers have used various theories and models to manage water supply and demand under different conditions, a comprehensive approach based on portfolio theory and the dynamics of urban water systems is missing in the literature. As mentioned before, portfolio theory applies to the selection of the optimized portfolio that provides reliable water supply services at different risk preferences. However, it does not decide on the time of increase in supply and protective measures during the planning period (Beyhaghi & Hawley 2013). Beyhaghi & Hawley (2013) argued that portfolio theory ignored the dynamics of the system in water resources management. Therefore, there was a need for the development of a mathematical model that could integrate the fundamental principles of portfolio theory with the critical properties of the physical water supply systems. To this end, a problem named the optimal sequencing of water supply options in water supply management has been formulated. In order to increase the complexity of optimal sequence plans, Beh et al. (2014) employed two approaches to determining the optimal sequence of water supply projects. Their multi-objective optimization model aimed to minimize the present value of cost and the amount of greenhouse gas emissions. Moreover, Herman et al. (2014) and Roach et al. (2016) assessed appropriateness and the performance of the decision-making methods with optimal sequencing in water resource management. They emphasized the uncertainties of supply and demand using a portfolio with robust planning to solve the demand growth problem.

Mulvey et al. (1995) introduced robust optimization and compared it with two traditional approaches, namely sensitivity analysis and stochastic linear programming. Afterwards, it was extended to real applications. Watkins & McKinney (1995) modeled uncertainty of water resources with different scenarios and found solutions for the model through robust optimization.

The robust optimization is an approach to the control of the uncertainty and cost involved in water resource problems. In order to avoid economic and social crisis in conditions of extreme drought, Mortazavi-Naeini et al. (2015) developed a model with Interval Linear Programming (ILP), Stochastic Robust Optimization (SRO), and Two-stage Stochastic Programming (TSP) techniques for solving the water resource allocation problem under different scenarios. They showed that a stochastic model with a robust optimization framework solved by a multi-objective evolutionary algorithm was suitable to cope with the complexity of urban bulk water systems for decision makers. To overcome the complexity posed by deep uncertainties such as climate change, Beh et al. (2017) developed a meta-model that made explicit consideration of robustness possible as an objective within a multi-objective optimization framework. They applied Artificial Neural Network (ANN) meta-models to the optimal sequencing of the urban water supply augmentation options in the water supply system. Robust water resources management under uncertainty was applied under an MCDM framework. Zhu et al. (2019) proposed a weight aggregation approach in an uncertain environment based on game theory with feasible weight space and combined stochastic multi-criteria acceptability analysis theory and grey relational analysis for robust water resources management. To improve the efficiency of resource allocation and due to the need for new infrastructure, Gold et al. (2019) incorporated regional water portfolio planning systems, including water transfer and regional demand management. They found that uncertainty could increase robustness conflicts in cooperative regional water portfolio planning.

For reducing the risk of water deficit in water resource management, Roach et al. (2016) compared the robust optimization and info-gap methods under a range of uncertain future scenarios by maximizing the robustness of long-term water supply and minimizing the associated costs of adaptation strategies. To solve optimization problems with uncertainty, Yanıkoğlu et al. (2019) reviewed adjustable robust optimization. Of note, some flexible decision variables can compare adjustable robust optimization with static robust optimization.

In Table 1, for a better view of the problem, some recent papers with case-based models are summarized and categorized for planning urban water supply and demand under uncertainty with regard to costs. In the first category, different types of decision variables, including continuous, binary, and integer, used in various mathematical formulations are provided. In the second category, different approaches such as Fuzzy Sets (FS) theory, Stochastic Programming (SP), and Robust Optimization (RO) techniques employed to model the uncertainties in the model parameters are given. The third category shows whether the models are single-period or multi-period. The optimization problems are brought in the fourth category based on the number of objectives as single-objective and multi-objective. In the fifth category, four groups of techniques for solving the problems, including exact methods (E), heuristic algorithms (H), meta-heuristic algorithms (MH), and simulation (S), are introduced. The sixth category determines whether a research study has used water supply, water demand, or both. In the seventh category, it is established whether a portfolio is selected to reduce the risk or not. The type of objective function (economic or per capita water) is determined in the eighth category. Finally, since all the research works surveyed are case-based, the locations for conducting them are shown in the final column of Table 1.

Table 1

Previous water management works

ReferenceModelUncertainty
Time period
Objective function
Solution technique
Options of portfolio
Portfolio selection
Element of objective function
Case study
FSSPROSingleMultipleSingleMultipleESHMHSupplyDemandYesNoEconomicalPer capita water
Beh Maier & Dandy (2015aMixed integer  •   •  •    • •  •  •  Southern Adelaide 
Mortazavi-Naeini et al. (2014)  Dynamic programming  •   •  •    • • •  • •  Sydney 
Herman et al. (2014)  Many-Objective Robust Decision Making (MORDM)  • •  •  •    • • • •  •  Region of North Carolina, USA 
Beh et al. (2015bMixed integer  •   •  •    • •  •  •  Southern Adelaide 
Kang & Lansey (2011)  Mixed integer  •  •  •  •   • •   • •  Arizona 
Leroux & Martin (2016)  Dynamic programming  •  •  •   •   •  •  •  Melbourne 
Leroux & Martin (2014)  Dynamic programming  •  •  •   •   •   • •  Melbourne 
Matrosov et al. (2015)  Mixed integer optimization  •  •   •    • •  •  •  London 
Mortazavi et al. (2012)  Mixed integer optimization  •  •   •    • •   • •  Sydney 
Rosenberg & Lund (2009)  Stochastic mixed integer  •   • •     • •   • •  Amman (Jordan) 
Vieira & Cunha (2016)  Simulated annealing nonlinear programming  •  •  •     • •   • •  Western Algarve 
Roach et al. (2016)  Robust optimization and info-gap methods  •  •  •     • • •  • •  United Kingdom 
Miao et al. (2014)  Interval–Fuzzy De Novo Programming (IFDNP) • •  •   •  •   • •  • •  • 
Kanta & Zechman (2014)  Agent-Based Modeling (ABM)  •  •  •   •   • •  • •  Arlington 
Beh et al. (2017)  Artificial Neural Network (ANN)  • •  •  •    • •   • •  Southern Adelaide 
Our work Mixed Integer Nonlinear Programming (MINLP)  • •  •  • •    • • •  • • Karaj 
ReferenceModelUncertainty
Time period
Objective function
Solution technique
Options of portfolio
Portfolio selection
Element of objective function
Case study
FSSPROSingleMultipleSingleMultipleESHMHSupplyDemandYesNoEconomicalPer capita water
Beh Maier & Dandy (2015aMixed integer  •   •  •    • •  •  •  Southern Adelaide 
Mortazavi-Naeini et al. (2014)  Dynamic programming  •   •  •    • • •  • •  Sydney 
Herman et al. (2014)  Many-Objective Robust Decision Making (MORDM)  • •  •  •    • • • •  •  Region of North Carolina, USA 
Beh et al. (2015bMixed integer  •   •  •    • •  •  •  Southern Adelaide 
Kang & Lansey (2011)  Mixed integer  •  •  •  •   • •   • •  Arizona 
Leroux & Martin (2016)  Dynamic programming  •  •  •   •   •  •  •  Melbourne 
Leroux & Martin (2014)  Dynamic programming  •  •  •   •   •   • •  Melbourne 
Matrosov et al. (2015)  Mixed integer optimization  •  •   •    • •  •  •  London 
Mortazavi et al. (2012)  Mixed integer optimization  •  •   •    • •   • •  Sydney 
Rosenberg & Lund (2009)  Stochastic mixed integer  •   • •     • •   • •  Amman (Jordan) 
Vieira & Cunha (2016)  Simulated annealing nonlinear programming  •  •  •     • •   • •  Western Algarve 
Roach et al. (2016)  Robust optimization and info-gap methods  •  •  •     • • •  • •  United Kingdom 
Miao et al. (2014)  Interval–Fuzzy De Novo Programming (IFDNP) • •  •   •  •   • •  • •  • 
Kanta & Zechman (2014)  Agent-Based Modeling (ABM)  •  •  •   •   • •  • •  Arlington 
Beh et al. (2017)  Artificial Neural Network (ANN)  • •  •  •    • •   • •  Southern Adelaide 
Our work Mixed Integer Nonlinear Programming (MINLP)  • •  •  • •    • • •  • • Karaj 

Comparing the last row of Table 1 with the works shown in the other rows, the main contributions of the current research are as follows:

  • (1)

    While a single economic criterion has been considered in most studies in the field of urban water management, in this research, two objective functions, namely economic and per capita water, are used.

  • (2)

    Robust changes in the results of urban water supply and demand are considered using a hybrid approach of stochastic and robust optimization.

  • (3)

    The Standardized Precipitation Evapotranspiration Index (SPEI) is used to generate different scenarios for the first time.

In managing urban water supply and demand, a risk-based approach with portfolio selection can identify uncertainties to an acceptable level and help managers in more accurate planning (Trindade 2019).

The features of the current paper and the need for conducting more in-depth research are explained in this section. The case study and the scenario tree are introduced in the next section. Then, the mathematical model will be provided. After proposing the approach, the results and a discussion on them are given. Finally, the conclusions and some suggestions for future research are presented.

CASE STUDY

Karaj is the capital city of the Alborz province of Iran and a suburb of Tehran. The population of the city is around 1.592 million (as recorded in the 2016 census) and its urban area (162 km2) is the fourth largest in Iran. Karaj's water resources are classified as surface water, groundwater, and dams, 23% of which are consumed in the urban sector, 3% in the industrial sector, and the highest percentage (74%) in the agricultural sector (Statistical Centre of Iran). The required data for the mathematical model were gathered for Karaj city. The ARIMA technique in the Minitab software was applied to the data in order to forecast the next 20 years.

As Table 2 shows, the population of Karaj city has grown significantly from 1956 to 2016. These figures are also used in the second objective function.

Table 2

Population growth of Karaj city

Year19561966197619861996200620112016
Population 14,526 44,243 137,926 275,100 940,968 1,377,450 1,424,187 1,592,492 
Growth rate – 11.1 11.4 6.9 12.3 3.8 3.1 2.3 
Year19561966197619861996200620112016
Population 14,526 44,243 137,926 275,100 940,968 1,377,450 1,424,187 1,592,492 
Growth rate – 11.1 11.4 6.9 12.3 3.8 3.1 2.3 

One of the essential parameters in determining the amount of water in a reservoir is the annual rainfall rate. Moreover, using the historical data and the time-series tools, precipitation rates of Karaj were forecast for the following years, as shown in Table 3. The ARIMA technique used to forecast the annual precipitation is described in Appendix C.

Table 3

Annual rainfall of Karaj city (in millimetres)

Year 1 2 3 4 5 6 7 8 9 10   
Rainfall 140 344 328 236 157 161 219 188 230 305   
Year 11 12 13 14 15 16 17 18 19 20   
Rainfall 314 158 209 228 422 213 170 303 268 365   
Year 21 22 23 24 25 26 27 28 29 30 31 32 
Rainfall 270 299 394 157 276 292 380 287 132 159 215 214 
Year 1 2 3 4 5 6 7 8 9 10   
Rainfall 140 344 328 236 157 161 219 188 230 305   
Year 11 12 13 14 15 16 17 18 19 20   
Rainfall 314 158 209 228 422 213 170 303 268 365   
Year 21 22 23 24 25 26 27 28 29 30 31 32 
Rainfall 270 299 394 157 276 292 380 287 132 159 215 214 

The forecasted annual rainfall is presented in Figure 1.

Figure 1

Forecasted annual rainfall (in millimetres).

Figure 1

Forecasted annual rainfall (in millimetres).

The water demand of Karaj city between 2002 and 2017 is reported in Table 4.

Table 4

Water demand of Karaj city (m3)

Year 2002 2003 2004 2005 2006 2007 2008 2009 
Demand 35,487,262 36,145,870 37,045,682 37,701,260 38,016,875 38,412,510 39,063,645 41,999,431 
Year 2010 2011 2012 2013 2014 2015 2016 2017 
Demand 42,535,690 42,093,417 41,559,029 40,848,990 39,820,762 41,276,902 41,590,646 44,566,511 
Year 2002 2003 2004 2005 2006 2007 2008 2009 
Demand 35,487,262 36,145,870 37,045,682 37,701,260 38,016,875 38,412,510 39,063,645 41,999,431 
Year 2010 2011 2012 2013 2014 2015 2016 2017 
Demand 42,535,690 42,093,417 41,559,029 40,848,990 39,820,762 41,276,902 41,590,646 44,566,511 

Scenario tree

In this section, the scenario tree generation method is described. According to the second category in Table 1, the uncertainty of hydrological parameters is stochastic. Therefore, the scenario tree will be used for the uncertainty. For this purpose, SPEI, a drought characteristic for analysis and comparison, is employed. Classifying drought as dry, normal, and wet according to the SPEI value is shown in Table 5 (Beguería et al. 2014).

Table 5

| Occurrence probabilities of the scenarios

Drought classDry (α)Normal (β)Wet (γ)
SPEI index value Index < −1.0 −1.0 ≤ Index ≤1.0 1.0 < Index 
Probability 0.54 0.10 0.36 
Drought classDry (α)Normal (β)Wet (γ)
SPEI index value Index < −1.0 −1.0 ≤ Index ≤1.0 1.0 < Index 
Probability 0.54 0.10 0.36 

The SPEI data from 1986 to 2015 were gathered for the Karaj water company. It should be noted that since long-term planning was the aim, all data were annual. Using the time-series module in the Minitab software, the forecasts were produced for the next 20 years. Moreover, the normal probability plot shows that there is no significant departure from normality.

Based on the predicted values of SPEI for the next 20 years, the occurrence probabilities for dry, wet, and normal years are given in Table 5. The occurrence probabilities of the scenarios are approximately equal to the frequencies of dry, normal, and wet conditions in the coming years. Accordingly, the following steps are taken to calculate the occurrence probabilities of dry, normal, and wet years given in Table 5:

  • The data for the past 20 years is collected.

  • The time-series technique is used to predict the next 20 years.

  • With the data for the past and the next 20 years, the frequencies of occurrence for dry, normal, and wet years are determined.

According to the papers in Table 1, the planning is designed for the next 20 years in four five-year intervals. On the other hand, the SPEI has three classifications to describe future drought characteristics. Thus, 81 scenarios are produced (Pingale Jat & Khare 2014). These scenarios are presented in Figure 2.

Figure 2

Scenario tree (cf. Table 5 and the first two columns of Appendix B).

Figure 2

Scenario tree (cf. Table 5 and the first two columns of Appendix B).

Mathematical model

The mathematical model in this section aims to minimize the cost and maximize the per capita water. It should achieve a target reliability level, minimize shortage in case of occurrence, and select new supply augmentation and demand management options in the outcome. Here, two types of risk are defined: the severity considered in the first objective function and the shortage frequency in Constraint (6) below. To reduce the risk and increase the returns, the portfolio selection method is used. Precipitation and streamflow are the main uncertain parameters on the supply side and population growth is an uncertain parameter that directly affects the demand for water. Due to the uncertainty involved in supply and demand in managing natural resources, a scenario-based stochastic optimization approach is utilized (Beyhaghi & Hawley 2013). In addition, due to the dynamic behavior of water resources, which requires sequential decision making, the results are expressed at each stage based on a multistage scenario tree. The planning horizon in this study is considered 20 years with four decision intervals of five years. Moreover, 81 possibilities are generated using the time-series data in the scenario tree.

The assumptions made for the mathematical model are:

  • (1)

    The first, second, and third options are the resources that currently supply urban water to Karaj city. Thus, no initial investment costs are assumed for these portfolios.

  • (2)

    Karaj city is the case study presented in this paper.

  • (3)

    Water shortage is a condition in which the urban water supply cannot fulfill the demand.

  • (4)

    30% of the precipitation evaporates.

  • (5)

    It is assumed that all water resources, streamflow, precipitation, groundwater, and water behind the dams are first gathered into a hypothetical reservoir and then distributed among urban consumers.

  • (6)

    The amount of supply and the shortage must meet the amount of demand.

  • (7)

    30% of the water is wasted when distributed through the pipelines.

In the following, the indices, the parameters, and the decision variables used to model the problems are defined.

The indicators are as follows:

i I is the number of portfolio options  
j J is the number of decision stages  
r M is the number of scenarios (r= 1,2,..,s,…, M
t T is the number of time periods 
i I is the number of portfolio options  
j J is the number of decision stages  
r M is the number of scenarios (r= 1,2,..,s,…, M
t T is the number of time periods 

The parameters are as follows:

 The occurrence probability of the scenario r that specifies the scenario tree 
 Fixed investment cost of option i in stage j under scenario r 
 The variable cost of option i under scenario r in period t 
 Penalty cost of shortage under scenario r in period t 
 The demand for urban water under scenario r in period t 
 Amount of streamflow under scenario r in period t 
 Precipitation deposited directly under scenario r in period t 
 Amount of released water for non-urban use under scenario r in period t 
 Amount of water evaporation under scenario r in period t 
 The lower limit for the (supply or demand) option i in stage j under scenario r in period t 
 The upper limit for the (supply or demand) option i in stage j under scenario r in period t 
 The population of Karaj city in period t 
 The risk level of the urban water shortage 
L A large number 
 The occurrence probability of the scenario r that specifies the scenario tree 
 Fixed investment cost of option i in stage j under scenario r 
 The variable cost of option i under scenario r in period t 
 Penalty cost of shortage under scenario r in period t 
 The demand for urban water under scenario r in period t 
 Amount of streamflow under scenario r in period t 
 Precipitation deposited directly under scenario r in period t 
 Amount of released water for non-urban use under scenario r in period t 
 Amount of water evaporation under scenario r in period t 
 The lower limit for the (supply or demand) option i in stage j under scenario r in period t 
 The upper limit for the (supply or demand) option i in stage j under scenario r in period t 
 The population of Karaj city in period t 
 The risk level of the urban water shortage 
L A large number 

Decision variables are as follows:

 1 – if supply augmentation or demand management measure of option i in stage j under scenario r is selected; 0 – otherwise 
 Amount of water from option i under scenario r in period t (it is explained in the following) 
 1 – if water shortage under scenario r in period t occurs; 0 – otherwise 
 Amount of water shortage under scenario r in period t 
 1 – if supply augmentation or demand management measure of option i in stage j under scenario r is selected; 0 – otherwise 
 Amount of water from option i under scenario r in period t (it is explained in the following) 
 1 – if water shortage under scenario r in period t occurs; 0 – otherwise 
 Amount of water shortage under scenario r in period t 

For identification of the decision variables, all potential portfolio components are defined. In this case, the candidate supply augmentation and demand management options should have geological (physical), political, environmental, economic, and social feasibility. Eleven portfolio components are selected to be in the portfolio. The amount of water for decision variables (options ) under scenario r in period t is defined as:

 Supply Groundwater (well) 
 Supply Taleghan dam 
 Supply Karaj dam (Amirkabir) 
 Supply Groundwater plain of Karaj 
 Supply Taleghan dam transmission line to Bilghan basin 
 Supply Surface water resources of the Karaj river 
 Supply Water harvesting from the Karaj river through Felman well excavation 
 Supply Water harvesting from the Kordan cone wells 
 Supply Assignment from the Chalus field or replacement with the Taleghan dam 
 Demand Replacing old infrastructures 
 Demand Training and advertising 
X {x1,r,t,x2,r,t, …, x11,r,t
 Supply Groundwater (well) 
 Supply Taleghan dam 
 Supply Karaj dam (Amirkabir) 
 Supply Groundwater plain of Karaj 
 Supply Taleghan dam transmission line to Bilghan basin 
 Supply Surface water resources of the Karaj river 
 Supply Water harvesting from the Karaj river through Felman well excavation 
 Supply Water harvesting from the Kordan cone wells 
 Supply Assignment from the Chalus field or replacement with the Taleghan dam 
 Demand Replacing old infrastructures 
 Demand Training and advertising 
X {x1,r,t,x2,r,t, …, x11,r,t
The mathematical formulation of the problem is as follows:
formula
(1)
formula
(2)
Subject to:
formula
(3)
formula
(4)
formula
(5)
formula
(6)
formula
(7)
formula
(8)
formula
(9)

The objective function (1) is mainly designed to minimize the fixed and variable costs and ensure security against water scarcity calculated as Min total cost = Fixed cost + Variable Cost + Penalty cost of shortage.

Fixed and variable costs are related to investment in water infrastructure and consumption reduction programs. Penalty cost is the monetary amount of vulnerability risk. According to a survey, people tend to pay approximately three times the current cost of urban water to avoid scarcity and shortages. This cost is considered as a fee for the penalty of urban water shortage . Objective function (2) maximizes the per capita supply of water for more social welfare. Constraint (3) addresses the balance of the total volume of water in the reservoir. It guarantees the balance between the amounts of reservoir water in two consecutive periods with the total amount of streamflow, precipitation, evaporation, non-income (non-urban) water use, and water demand. The streamflow and precipitation are resources of the reservoir gathered from different stations of the Adaran, Mowrud, Kalvan, Nasht-e Rud, Velayat Rud, and Karaj rivers. Evaporation is an unwanted cause of water waste. However, over time, it will return to the water cycle. Part of the water entering the reservoir is used for non-urban purposes, which must be deducted from the total incoming water. Demand is real consumption. Constraint (4) imposes the lower and upper bounds on the capacity of options for the delivery of water from the supply resources. Constraint (5) enforces a balance of water supply with water demand and shortage. When the available supply is sufficient to satisfy demand, gr,t is 0. Constraint (6) ensures the minimum target reliability level and is a frequency measure of water shortage risk. Finally, constraint (7) shows that the amount of shortage accrues with the number of plausible shortages. Since the variable is binary and the amount of shortage depends on its occurrence, this constraint has been added. Relationships (8) and (9) show the status of the decision variables.

PROPOSED APPROACH BASED ON HYBRID ROBUST STOCHASTIC OPTIMIZATION

To solve the complex nonlinear programming problem modeled in the previous section, three major steps are taken in this section. Figure 3 demonstrates an overview of the solution method. In Step 0, shown in Figure 3, the second objective function contributes to the maximization of the amount of assigned water to each person in the problem, which is crucial and should be included to make it a bi-objective optimization problem. The first and second objective functions conflict and should reach a trade-off. This justifies the use of Pareto optimal solutions. Reducing costs leads to a reduction in the allocated water; the former is desirable and the latter is not.

Figure 3

Overview of the proposed approach.

Figure 3

Overview of the proposed approach.

In Step 1 of Figure 3, the first nonlinear objective function is linearized using the following definition and new constraints:
formula
(10)
formula
(11)
formula
(12)
formula
(13)
Then, the model becomes:
formula
(14)
formula
(15)

Subject to:

Constraints (3)–(7);
formula
(16)
formula
(17)
formula
(18)
formula
(19)
formula
(20)
Exact solution methods are not obtainable by most of the solution techniques presented in the literature with multi-objective functions (Table 1, fifth category). The main exact solution methods are the weighted sum method and the ε-constraint. Our proposed solution approach is exact and the scales of the conflicting objective functions are different. While cost minimization is the main objective in most of the studies, the first objective function with higher preference is assumed as the objective function and the second objective is converted to a constraint in this study. Then, the ε-constraint method to solve the multi-objective optimization problems is used (Allaoui et al. 2018). This method keeps only one of the objectives and restricts the rest within user-specific values (Step 2, Figure 3).
formula
(21)
Subject to:
formula
(22)
constraints (3)–(7), (16)–(20).

The new parameters for the robust formulation of the problem are as follows. Most parameters are described in Appendix A.

 The total amount of demand is determined by the standard consumption in different scenarios at different times 
 Nominal total demand in period t under scenario r 
 Nominal excessive consumption change that occurs in different scenarios 
 Positive deviation percentages from the nominal scenarios 
 Aggregated scaled deviation of certain parameters in constraint i 
 Definitions applied to the polyhedral uncertainty sets of the total demand 
 Set of surplus and shortage quantities in demand estimation 
 Waste cost per unit of excess supply of water (due to the excess water supply compared with the forecast or for various other reasons, water may be wasted in the transmission network and not used properly, in which case we will face the cost of waste) 
 Penalty cost per unit of uncollected demand (the cost of water supply is less than expected, which occurs in the transmission network for various reasons and is considered a kind of penalty) 
 Modified objective function value for scenario r 
 The total amount of demand is determined by the standard consumption in different scenarios at different times 
 Nominal total demand in period t under scenario r 
 Nominal excessive consumption change that occurs in different scenarios 
 Positive deviation percentages from the nominal scenarios 
 Aggregated scaled deviation of certain parameters in constraint i 
 Definitions applied to the polyhedral uncertainty sets of the total demand 
 Set of surplus and shortage quantities in demand estimation 
 Waste cost per unit of excess supply of water (due to the excess water supply compared with the forecast or for various other reasons, water may be wasted in the transmission network and not used properly, in which case we will face the cost of waste) 
 Penalty cost per unit of uncollected demand (the cost of water supply is less than expected, which occurs in the transmission network for various reasons and is considered a kind of penalty) 
 Modified objective function value for scenario r 
According to Appendix A, the ultimately proposed hybrid stochastic robust formulation of the problem is shown in (23)–(36) (Step 3, Figure 3):
formula
(23)
Subject to constraints (4)–(7), (16)–(20), (22),
formula
(24)
formula
(25)
formula
(26)
formula
(27)
formula
(28)
formula
(29)
formula
(30)
formula
(31)
formula
(32)
formula
(33)
formula
(34)
formula
(35)
formula
(36)

RESULTS AND DISCUSSION

In this section, the Benders decomposition method is used to solve the urban water portfolio planning problem for Karaj city, robustly modeled by the solution approach (Step 4, Figure 3). Figure 4 shows the steps taken in the Benders decomposition algorithm. The results for different options at different decision-making stages and scenarios are obtained by GAMS 22.9, which uses the CPLEX solver, on a laptop with Core i5 2.5 GHz CPU and 4.0 GB of RAM. The Benders decomposition in GAMS is employed, because this optimization software can handle loops, which is essential in the algorithm.

Figure 4

Benders decomposition algorithm.

Figure 4

Benders decomposition algorithm.

The amounts of water for decision variables (some given as examples) are presented in Table 6 over different years and different scenarios in millions of cubic metres.

Table 6

Amounts of water (decision variables 1–6) at different decision stages

YearOption 1
Option 2
Option 3
Option 4
Option 5
Option 6
DryNormalWetDryNormalWetDryNormalWetDryNormalWetDryNormalWetDryNormalWet
2020 – – – – – – – – 
2021 – – – – – – – – 
2022 – – – – – – – – 
2023 – – – – – – – – 
2024 – – – – – – – – 
2025 – – – – – – – – 
2026 – – – – – – – – 
2027 – – – – – – – – 
2028 – – – – – – – – 
2029 – – – – – – – – 
2030 – – – – – – – – 
2031 – – – – – 30 – – 
2032 – – – – – 30 – – 
2033 – – – – – 30 – – 
2034 – – – – – 30 – – 
2035 – – – – – 30 – – 
2036 – – – – – 30 – – 
2037 – – – – – 40 40 – 
2038 – – – – 40 40 – 
2039 – – – – 40 40 – 
2040 – – – – 40 40 – 
YearOption 1
Option 2
Option 3
Option 4
Option 5
Option 6
DryNormalWetDryNormalWetDryNormalWetDryNormalWetDryNormalWetDryNormalWet
2020 – – – – – – – – 
2021 – – – – – – – – 
2022 – – – – – – – – 
2023 – – – – – – – – 
2024 – – – – – – – – 
2025 – – – – – – – – 
2026 – – – – – – – – 
2027 – – – – – – – – 
2028 – – – – – – – – 
2029 – – – – – – – – 
2030 – – – – – – – – 
2031 – – – – – 30 – – 
2032 – – – – – 30 – – 
2033 – – – – – 30 – – 
2034 – – – – – 30 – – 
2035 – – – – – 30 – – 
2036 – – – – – 30 – – 
2037 – – – – – 40 40 – 
2038 – – – – 40 40 – 
2039 – – – – 40 40 – 
2040 – – – – 40 40 – 

The results in Table 6 indicate that the first three current options (well, Taleghan dam, Karaj dam) in the urban water supply network of Karaj are considered for the subsequent stages in 20 years with the available water of 1 and 2 million cubic metres each for dry, normal, and wet conditions. The results also show that the last two options (replacing old infrastructures, training and advertising), which are managerial and educational, are planned in a portfolio in all decision-making stages and scenarios due to the insignificant investment cost compared with others.

Option 4 (groundwater plain of Karaj) is reported for all the dry years and the last three years of the normal scenario. While option 5 (Taleghan dam transmission line to Bilghan basin) is not chosen, the sixth option (surface water resources of the Karaj river) is also included in the last years in the dry and normal year scenarios to cover the demand. It is necessary to invest in this option in a dry year from 2031 and a normal year from 2037. This option is not required in the wet year scenario.

The amount of shortage over the years is zero due to high costs on the one hand and the decline in population growth in the coming years, which will significantly reduce the demand intensity, on the other hand. The optimal values of the two objective functions are shown in Table 7.

Table 7

Optimal values of the objective functions (the costs are expressed in Iranian Rials (IRR))

Objective function 1Objective function 2
Optimal cost of using different options (IRR) Optimal per capita water (m3/person) 
248,653,800,000,000 26,003 
262,467,900,000,000 27,447 
276,282,000,000,000 28,892 
290,096,100,000,000 30,337 
303,910,200,000,000 31,781 
Objective function 1Objective function 2
Optimal cost of using different options (IRR) Optimal per capita water (m3/person) 
248,653,800,000,000 26,003 
262,467,900,000,000 27,447 
276,282,000,000,000 28,892 
290,096,100,000,000 30,337 
303,910,200,000,000 31,781 

One of the parameters defined in the model (in constraint (6)) is . At the beginning of the model's solution for determining the water portfolio of Karaj city, was set to 0.001. However, due to the lack of shortage in the Karaj water supply, changing the amount of did not affect the solution.

Changing the amount of demand in the first objective function is highly influential. For instance, the cost obtained by the first objective function when the demand is in the range between 50% reduction and 100% increase is depicted in Figure 5.

Figure 5

Effect of demand change on the cost of the first objective function.

Figure 5

Effect of demand change on the cost of the first objective function.

As mentioned, one of the high costs in water supply in the water transmission network is the cost of wastewater. This cost occurs in the transmission network for several reasons including excess supply and water pipe wear on transmission and lack of knowledge on how to use it properly. This cost can be reduced in the transmission network by using various methods, e.g., increasing consumer awareness, but it never reaches zero. Therefore, it seems necessary to perform sensitivity analysis for this parameter. In Figure 6, with different cost loss values, the effect of increasing the total cost is investigated. By increasing the percentage for this cost from 0% to 25%, the total cost value increases from 0% to 40% and remains almost constant in a low deviation domain.

The waste cost per unit of excess supply of water is one of the influential parameters on the extent of uncertainty. This parameter, as well as other similar parameters, affects the demand in robust conditions. For instance, Figure 6 shows how the cost affects the value of when it increases to 25% by a rise in cost to 40% and remains constant.

Figure 6

Effect of cost on the estimation of urban water demand 〖SC〗^d.

Figure 6

Effect of cost on the estimation of urban water demand 〖SC〗^d.

The average cost of the first objective function in various scenarios is presented in Figure 7.

Figure 7

Average cost of the first objective function in various scenarios.

Figure 7

Average cost of the first objective function in various scenarios.

Figure 8

Pareto front.

Figure 8

Pareto front.

All the related results are given in Appendix B. They indicate that the average cost of the model for the first objective function in various scenarios is 3.39 × 1012. Meanwhile, the average per capita water is predicted at 10,149 during the 20 years.

In many cases, including this one, the objective functions defined in the multi-objective optimization problem have a trade-off, justifying the use of Pareto optimal solutions. According to all the previous studies (especially those given in Table 1), in such problems, cost minimization is greatly important. Meanwhile, in these studies, the ε-constraint method is used and the first objective function is selected because the method gives more importance to one objective function (Allaoui et al. 2018). In Figure 8, the ε-constraint method is used to find the Pareto front.

As shown in Figure 8, a Pareto front is represented in the space of objective functions. This curve shows that the values of the objective functions can be obtained in the form of the Pareto optimal front. Thus, all the points on the curve are indifferent. Moreover, a decision maker can choose the points according to their preference and the Pareto optimal solutions cannot be unique.

CONCLUSION AND RECOMMENDATIONS FOR FUTURE RESEARCH

Due to the uncertainty involved, the planning problem of urban water supply and demand simultaneously has not been addressed in the literature. This paper aimed to focus on supply and demand portfolios in uncertain conditions. With this aim, a bi-objective stochastic optimization model was first developed to minimize the total cost and to maximize the per capita water. Then, the first objective function was linearized. Next, the bi-objective model was converted to a single-objective model using the ε-constraint method. After that, a robust counterpart of the model was presented. Finally, the robust model was solved using the Benders decomposition method available in the GAMS software.

In this paper, major predictions were made regarding population growth, water demand, and fixed and variable investment costs for the water supply in different options. On the other hand, the average interest rate was 20%. Under three main scenarios for each stage, which led to 81 scenarios for the next four five-year periods, the reduction in precipitation was estimated. The results showed that mostly the first to the third options, which are currently supplying Karaj water, were selected with demand options for the water supply during the next 20 years.

The following managerial points are recommended based on the results of the current research:

  • (1)

    The growth of the Karaj population over the next 20 years and a decline in precipitation will reduce the water inventory of the reservoirs (total urban water). Therefore, urban water managers should plan to reduce the amount of water evaporation in various water resources, which is currently approximately 30% of the precipitation.

  • (2)

    Due to the relatively high deviation between the occurrence of different scenarios and the results, urban water managers should carefully monitor the exact amount of precipitation, streamflow, and urban water demand to accurately define the scenarios.

  • (3)

    Demand management tools are preventive and should always be considered.

The following topics are recommended for future research:

  • In this paper, the subject of urban water supply was considered uniformly for the whole population and not for clustered populations. Data mining methods can be used to cluster the population and estimate different consumptions for the clustered populations.

  • While the urban water supply was strategically planned for the next 20 years in five-year periods, medium and tactical planning for each year can be studied.

  • While in this paper, evaporation was defined by the experts as a percentage of precipitation, it can be determined based on the weather conditions implemented in the model.

DATA AVAILABILITY STATEMENT

Data cannot be made publicly available; readers should contact the corresponding author for details.

REFERENCES

Allaoui
H.
Guo
Y.
Choudhary
A.
Bloemhof
J.
2018
Sustainable agro-food supply chain design using two-stage hybrid multi-objective decision-making approach
.
Computers and Operations Research
89
,
369
384
.
https://doi.org/10.1016/j.cor.2016.10.012
.
Beguería
S.
Vicente-Serrano
S. M.
Reig
F.
Latorre
B.
2014
Standardized precipitation evapotranspiration index (SPEI) revisited: parameter fitting, evapotranspiration models, tools, datasets and drought monitoring
.
International Journal of Climatology
34
(
10
),
3001
3023
.
https://doi.org/10.1002/joc.3887
.
Beh
E. H. Y.
Dandy
G. C.
Maier
H. R.
Paton
F. L.
2014
Optimal sequencing of water supply options at the regional scale incorporating alternative water supply sources and multiple objectives
.
Environmental Modelling and Software
53
,
137
153
.
https://doi.org/10.1016/j.envsoft.2013.11.004
.
Beh
E. H. Y.
Maier
H. R.
Dandy
G. C.
2015a
Adaptive, multiobjective optimal sequencing approach for urban water supply augmentation under deep uncertainty
.
Water Resources Research
51
(
3
),
1529
1551
.
https://doi.org/10.1002/2014WR016254
.
Beh
E. H. Y.
Maier
H. R.
Dandy
G. C.
2015b
Scenario driven optimal sequencing under deep uncertainty
.
Environmental Modelling and Software
68
,
181
195
.
https://doi.org/10.1016/j.envsoft.2015.02.006
.
Beh
E. H. Y.
Zheng
F.
Dandy
G. C.
Maier
H. R.
Kapelan
Z.
2017
Robust optimization of water infrastructure planning under deep uncertainty using metamodels
.
Environmental Modelling and Software
93
,
92
105
.
https://doi.org/10.1016/j.envsoft.2017.03.013
.
Beuhler
M.
2006
Application of modern financial portfolio theory to water resource portfolios
.
Water Science and Technology: Water Supply
6
(
5
),
35
41
.
https://doi.org/10.2166/ws.2006.828
.
Beyhaghi
M.
Hawley
J. P.
2013
Modern portfolio theory and risk management: assumptions and unintended consequences
.
Journal of Sustainable Finance & Investment
3
(
1
),
17
37
.
Gold
D. F.
Reed
P. M.
Trindade
B. C.
Characklis
G. W.
2019
Identifying actionable compromises: navigating multi-city robustness conflicts to discover cooperative safe operating spaces for regional water supply portfolios
.
Water Resources Research
55
(
11
),
9024
9050
.
https://doi.org/10.1029/2019WR025462
.
Herman
J. D.
Zeff
H. B.
Reed
P. M.
Characklis
G. W.
2014
Beyond optimality: multistakeholder robustness tradeoffs for regional water portfolio planning under deep uncertainty
.
Water Resources Research
50
(
10
),
7692
7713
.
https://doi.org/10.1002/2014WR015338
.
Kang
D.
Lansey
K.
2011
A scenario-based optimization model for water supply system planning
. In:
World Environmental and Water Resources Congress 2011: Bearing Knowledge for Sustainability
(R. E. Beighley II & M. W. Killgore, eds),
ASCE
,
Reston, VA, USA
, pp.
146
155
.
Kanta
L.
Zechman
E.
2014
Complex adaptive systems framework to assess supply-side and demand-side management for urban water resources
.
Journal of Water Resources Planning and Management
140
,
75
85
.
https://doi.org/10.1061/(ASCE)WR.1943-5452.0000301
.
Leroux
A. D.
Martin
V. L.
2014
Optimal portfolio management of urban water
. In:
Australian Agricultural and Resource Economics Society Conference 2014
,
4–7 February, Port Macquarie, Australia
.
Leroux
A. D.
Martin
V. L.
2016
Hedging supply risks: an optimal water portfolio
.
American Journal of Agricultural Economics
98
(
1
),
276
296
.
https://doi.org/10.1093/ajae/aav014
.
Markowitz
H.
1952
Portfolio selection
.
The Journal of Finance
7
(
1
),
77
91
.
https://doi.org/10.2307/2975974
.
Matrosov
E. S.
Huskova
I.
Kasprzyk
J. R.
Harou
J. J.
Lambert
C.
Reed
P. M.
2015
Many-objective optimization and visual analytics reveal key trade-offs for London's water supply
.
Journal of Hydrology
531
,
1040
1053
.
https://doi.org/10.1016/j.jhydrol.2015.11.003
.
Miao
D. Y.
Huang
W. W.
Li
Y. P.
Yang
Z. F.
2014
Planning water resources systems under uncertainty using an interval-fuzzy De Novo programming method
.
Journal of Environmental Informatics
24
(
1
),
11
23
.
https://doi.org/10.3808/jei.201400277
.
Morgan
E. A.
Torabi
E.
Dedekorkut-Howes
A.
2020
Responding to change: lessons from water management for metropolitan governance
.
Australian Planner
56
(
2
),
125
133
.
https://doi.org/10.1080/07293682.2020.1742171
.
Mortazavi-Naeini
M.
Kuczera
G.
Cui
L.
2014
Application of multiobjective optimization to scheduling capacity expansion of urban water resource systems
.
Water Resources Research
50
(
6
),
4624
4642
.
https://doi.org/10.1002/2013WR014569
.
Mortazavi
M.
Kuczera
G.
Cui
L.
2012
Multiobjective optimization of urban water resources: moving toward more practical solutions
.
Water Resources Research
48
(
3
),
W03514
.
https://doi.org/10.1029/2011WR010866
.
Mortazavi-Naeini
M.
Kuczera
G.
Kiem
A. S.
Cui
L.
Henley
B.
Berghout
B.
Turner
E.
2015
Robust optimization to secure urban bulk water supply against extreme drought and uncertain climate change
.
Environmental Modelling and Software
69
,
437
451
.
https://doi.org/10.1016/j.envsoft.2015.02.021
.
Mulvey
J. M.
Vanderbei
R. J.
Zenios
S. A.
1995
Robust optimization of large-scale systems
.
Operations Research
43
(
2
),
264
281
.
https://doi.org/10.1287/opre.43.2.264
.
Paton
F. L.
Dandy
G. C.
Maier
H. R.
2014
Integrated framework for assessing urban water supply security of systems with non-traditional sources under climate change
.
Environmental Modelling and Software
60
,
302
319
.
https://doi.org/10.1016/j.envsoft.2014.06.018
.
Pingale
S. M.
Jat
M. K.
Khare
D.
2014
Integrated urban water management modeling under climate change scenarios
.
Resources, Conservation and Recycling
83
,
176
189
.
https://doi.org/10.1016/j.resconrec.2013.10.006
.
Roach
T.
Kapelan
Z.
Ledbetter
R.
Ledbetter
M.
2016
Comparison of robust optimization and info-gap methods for water resource management under deep uncertainty
.
Journal of Water Resources Planning and Management
142
(
9
),
04016028
.
https://doi.org/10.1061/(ASCE)WR.1943-5452.0000660
.
Rosenberg
D. E.
Lund
J. R.
2009
Modeling integrated decisions for a municipal water system with recourse and uncertainties: Amman, Jordan
.
Water Resources Management
23
,
85
.
https://doi.org/10.1007/s11269-008-9266-4
.
Trindade
B. C.
2019
Advancing Regional Water Supply Portfolio Management and Cooperative Infrastructure Investment Pathways under Deep Uncertainty
.
PhD thesis, Cornell University
,
Ithaca, NY, USA
.
https://doi.org/.1037//0033-2909.I26.1.78
Vieira
J.
Cunha
M.
2016
Systemic approach for the capacity expansion of multisource water-supply systems under uncertainty
.
Journal of Water Resources Planning and Management
142
(
10
),
04016034
.
https://doi.org/10.1061/(ASCE)WR.1943-5452.0000668
.
Watkins
D. W.
Jr.
McKinney
D. C.
1995
Robust optimization for incorporating risk and uncertainty in sustainable water resources planning
.
IAHS Publication
231
,
225
232
.
Yanıkoğlu
I.
Gorissen
B. L.
den Hertog
D.
2019
A survey of adjustable robust optimization
.
European Journal of Operational Research
277
,
799
813
.
https://doi.org/10.1016/j.ejor.2018.08.031
.
Zhu
F.
Zhong
P.
Cao
Q.
Chen
J.
Sun
Y.
Fu
J.
2019
A stochastic multi-criteria decision making framework for robust water resources management under uncertainty
.
Journal of Hydrology
576
,
287
298
.
https://doi.org/10.1016/j.jhydrol.2019.06.049
.

Supplementary data