**1. Introduction**

Existing statistics state that almost 55% of the population of the world currently resides in urban regions [1,2], with estimates that this rate will increase to 70% by 2050 [3]. As this progression towards greater urban centres continues to increase, a need has emerged to find ways for supporting this growth in a sustainable manner. Furthermore, there is the challenge of dealing with the pollution levels that result from exacerbated activities in cities [4]. Along with the surging rates of urbanisation and pollution, the world has also experienced a breakthrough in the use of technologies, specifically those related to information and communication technologies (ICT) [5]. Updates in connectivity between various electronic platforms has led to the development of the Internet of Things (IoT), which is based on networks formed between physical devices and appliances to allow data transfer and exchange for enhanced operations [6].

The integration of ICT and IoT is thought to lead to an enhanced system for the management of cities. As a result, the notion of smart cities has arisen in response to the need for sustainable cities that can accommodate the growing population numbers, hence enhancing cities' liveability and the wellness and living standards of citizens [7]. Even though universal agreement on a specific definition of a 'smart city' is still lacking, its main domain lies in the use of information and ICT in sectors such as infrastructure, buildings and energy [8]. In particular, concepts from ICT and IoT are being increasingly reflected in the operations of existing cities, resulting in an interrelated platform between large numbers of citizens, transport networks, services and urban assets and facilities [9].

Decision-making in planning and operations of smart cities needs to be structured around two main considerations, namely strategic and tactical decisions [10]. Yet, current emphasis in the literature is on the technical interfaces making up the various data-exchange enabling platforms placed within the cities, with little emphasis on the strategic and tactical urban planning aspects of smart cities. Within the area of strategic decision-making in city planning, zoning of the city and the location of its operating facilities, including schools, hospitals and so forth, are both of significant relevance [11]. An appropriate selection of zone clusters and the subsequent selection of locations to place buildings in form the main components that are involved in city design. In cities, the positioning of new buildings leads to the generation of traffic demand in the existing network structure [12]. This causes additional traffic loadings on the existing network, and if not well planned for, can result in major transportation delays to network users. Traffic congestion is thought to result in over \$121 billion in losses [13] and can increase the amount of carbon emissions from traffic by more than 53% [14]. Another important strategic consideration is the development of the underlying travel network of the region. Specifically, the development of the transport network will be based on the capacities required to handle the initiated travel on the transport networks, while the operations of the network will in turn be associated with the established capacity of the network, along with the traffic loading patterns on the links (roads) of the network. Decisions related to the transport network are determined based on population rates and estimated travel via the various transport modes utilised in the region [15]. As a result, the zoning of the city will have a direct impact on the locations available for the facilities required to service the underlying population, which in turn would also impact the traffic and operation of the transportation network [16]. It is vital to thus integrate the decision-making that is involved in the zoning, facility location and transport network capacity design of the underlying smart city.

The attention in this article is directed towards the concept of location planning in smart cities. The proposed approach can be divided into three main areas: (i) the need for establishing a framework for the construction of a smart city from scratch, where zoning and land use need to be specified; (ii) the location of buildings in smart cities and the investment decisions made regarding the expansion of the existing road network structure and capacity, which involves the consideration of attributes that influence the decision of positioning buildings such as schools, hospitals and offices; and (iii) the determination of the resulting impacts caused by such location decisions based on the triple bottom-line of sustainability, via formulation of appropriate social, environmental and economic cost objective functions. As a result, three main decisions, which form the essence of urban planning and design in smart cities are targeted: namely, the decisions made regarding the allocation of zones and the assignment of buildings to locations in the region, the expansion decisions related to the road structure of the city and the expansion of the capacity of existing links in the network (if one already exists).

In this paper, a range of mathematical optimisation problems are integrated, including the clustering problem [17], the assignment problem [18], the facility location problem [19] and the urban traffic design problem [12], in order to model key strategic decisions in smart city design and planning. The work proposed herein is expected to form an integral component of the urban design of smart cities. In particular, such a framework will find applicability in prospective smart city planning designs, to ensure a sustainable and integrated city structure where buildings and road networks are strategically planned for. The framework can be used both for new smart city development and for decisions to be made within an existing smart city and which impact the urban design morphology of the existing city.

Remainder of this article is divided as follows: the next section provides a review of the literature on smart city planning in terms of zoning, facility location and transport network design. The proposed mathematical optimisation framework for strategic planning of smart cities is then presented. Following that, the algorithm and formulations developed are outlined. A numerical

example of a case project is used to validate the proposed framework. Concluding remarks are provided at the end.

#### **2. Literature Review**

A significant number of research studies examine the concept of smart city and its use for sustainable urban planning. The literature reviewed herein falls under, but is not limited to, one of the following fields that are imperative building blocks in smart cities: energy management, city structure zoning and location, intelligent transport systems and smart infrastructure.

Key aspects regarding IoT and its use in smart cities were reviewed in [20]. A bibliometric analysis in [21] examines frequent city categorisation, including smart cities, used for sustainable urbanisation. A comprehensive review was conducted in [22] for energy management and sustainable planning in smart cities [23]. Spatiotemporal forecasting methods, which exploit time series data from various locations within the context of smart cities, and their applications for smart city transport and building management were evaluated in [24].

Several studies focused on energy management in smart cities [25]. Wojnicki and Kotulski [26] proposed an outdoor lighting control system for smart cities. An activity-aware system to automate building systems in smart cities was developed in [27]. A framework to optimise energy management on smart campuses was proposed in [28]. A heating and cooling modelling system was proposed for minimising electricity consumption in smart cities [29]. A smart city architecture was developed in [30] for addressing challenges in smart grid distribution. A comparative assessment of smart energy systems to ensure sustainable, clean and reliable energy in smart cities is found in [31].

In terms of location optimisation, the authors of [32] proposed an optimisation approach that relies on integrating geographic information systems (GIS) with a fuzzy-analytical hierarchy process (FAHP) for choosing suitable wind farm sites. Impacts of location choices made on wind turbine operations were discussed in [33]. In the context of infrastructure planning for smart cities, the authors of [34] developed a city navigation approach for electric vehicles, where locations of charging stations were assumed to be unknown. A business model tool for smart cities to facilitate undertaking strategic decision-making promptly was proposed in [35]. In [36], a localisation-based key management system for meter data encryption in smart cities was proposed. The utilisation of smart parking lots in smart cities was investigated in [37].

In terms of urban planning and structuring, an integrated model which considers land use, transportation and energy systems for future smart cities was presented in [38]. A framework for a smart city in China, which focuses on use of big data for infrastructure city planning and management, was developed by the authors of [39]. Use of smart technology for enhancing the sustainability of the construction sector through targeting demolition waste was discussed in [40]. Even though an investigation on algorithms deployed for sustainable transport policy in cities has been carried out, as detailed in [41], there is no link that has been developed to account for location decisions of new buildings and their impacts on the smart city urban design structure. New ways to achieve systematic-based solutions that augment the process adopted in urban design and planning, and which can lead to future viability and prosperity in metropolitan regions, need to be developed.

As is apparent, there is little focus on developing mathematical optimisation models that integrate location and transportation decisions for use in urban design of smart cities. In particular, there is an apparent lack in studies that focus on addressing operational research problems that are relevant to the strategic planning of urban areas. This is especially imperative in urban design of smart cities, where emphasis on the interconnectivity of key features within the city environment, including transportation systems and adjoining city layout, is highly regarded. As a result, in this work, the aspect within smart city design which will be examined refers to an automated and systematic urban planning procedure that relies on the use of mathematical optimisation frameworks. Such mathematical frameworks can be utilised for a range of applications in smart cities, including intelligent transport planning systems, intelligent energy monitoring and delivery and smart urban design approaches. The main contributions of this study are as follows: (i) the development of a mathematical method for dividing the proposed smart city region into zonal clusters; (ii) formulation of an assignment problem for land-use selection in zonal clusters; and (iii) development of a mathematical framework for enhancing the sustainable planning of location decisions made regarding building placement, and for measuring their impact on the routing of traffic within the smart city, through automated infrastructure investment decisions. The research presented herein thus integrates strategic aspects of location planning and traffic assignment involved in the urban design of smart cities. The next section outlines the main components making up the developed framework.

#### **3. Smart City Zoning, Facility Location and Transport Network Planning Framework**

The main motivation behind the framework proposed in this study is to ensure the efficient planning and design of city zoning, building location and transport network of smart cities, while accounting for environmental, social and economic considerations. As was previously discussed, the layout and location planning decision in smart cities considers two main planning aspects, namely strategic and operational planning; this is displayed in Figure 1. As can be seen, in terms of the strategic planning decisions, the main variables that need to be modelled in the proposed framework are the zoning of the region, the link expansion variables on the existing network, network extension through addition of new links, and the positioning of buildings. Apart from future population growth, which creates a slight increase in the demand induced in the regions, the focus in the developed framework is mostly on traffic demand generated by the placement of buildings.

**Figure 1.** Some of the operational research problems that can be considered when planning for urban design.

An outline of the proposed framework in this paper is shown in Figure 2. Three main decisions, which form the essence of urban planning and design in smart cities are targeted: namely, the decisions made regarding allocation of zones and the assignment of buildings to locations in the region, the expansion decisions related to the road structure of the city and the expansion of the capacity of existing links in the network (if one already exists). The main concept introduced via the developed framework is the vital integration of all three decisions into a single model that simultaneously optimises the decision-making process involved.

The strategic decision for smart cities starts with the division of the region into zones, assuming the subject is a new city that requires zoning. This step also involves determining the land-use patterns in the region. In the case that an already existing smart city is considered, then the zoning procedure can be neglected, and the existing zonal configuration can be adopted instead. The second step involves the positioning of the buildings within the zones defined and in accordance with the allocated land-use patterns. Examples of buildings that need to be located include offices, retail shops, hospitals and schools. The third step is that related to infrastructure development and expansion. Such decisions will be influenced by the previous two steps and so it is necessary to assimilate both decisions together in a fashion that permits the translations of the impacts that the zoning and location decisions have on the infrastructure investment decisions made by the planners.

**Figure 2.** Proposed framework for planning in smart cities.

Given that a chief consideration in smart cities is ensuring effective mobility and robust decision-making for improving the transport of goods and people, developing an approach that can incorporate a forward-looking method for assignment of traffic based on newly introduced buildings is thus imperative. To enable this to happen, a multi-objective optimisation model is developed, based on a bilevel structure [42]. The bilevel structure is needed in order to model the decision spaces of the two main decision-makers in the model: namely, the urban planners and the transportation network users. The importance of generating sustainable solutions that target the triple bottom-line of sustainability is also accounted for through considering objective functions in the optimisation model developed, with focus on environmental, social and economic impacts of the locations and infrastructure decisions made.

The model can then be adapted to continue to be utilised for the strategic decisions to be made within the smart city, whenever a change in the structure of the city is induced. The change that is emphasised in the framework is related to the introduction of a new zone or building within the region. As shown in Figure 2, the procedure loops back to the optimisation model, whose associated parameters are updated in response to the induced changes, and a new solution is generated. Otherwise, if no change is induced, the algorithm ceases.

The proposed framework can also be linked to other automated systems that rely on the use of ICT in daily management of the city, such as online estimation of origin–destination (OD) matrices within the city for enhanced traffic assignment and real-time traffic state estimation and updates [43]. The work in this article is specifically targeted towards enhancing the intelligence of the transport systems through considering the impacts of newly positioned buildings on the underlying network.

#### **4. Mathematical Optimisation Models**

In this section, the mathematical formulations that are integrated in the proposed framework are outlined. The mathematical optimisation models can be divided into three main types: the first is associated with the zone division and clustering in the region, the second relates to the assignment of land-use patterns to the zone clusters and the third is related to the location of buildings and traffic assignment to the underlying network, based on infrastructure decisions made in the region. Notations adopted in the proposed mathematical models are given in Nomenclature.

#### *4.1. Zoning Regions*

Assuming an input of free land dispersed in the region is provided by the decision-maker (urban planner), the first step in zoning a smart city involves clustering the land into discrete zones. This step will involve the use of a clustering algorithm in order to yield a layout of area nodes clustered into zones. The second step involves assigning a land-use zone to each cluster created via the clustering algorithm. Each of these steps will now be explained.

#### 4.1.1. Area Clustering

Each available area is clustered into a set of zones via the use of a clustering algorithm. In this study, principles from the k-means clustering approach are adopted [44]. The algorithm is summarised in Figure 3.

**Figure 3.** Steps involved in the k-means clustering approach adopted for zoning of the region. (**A**) Nodes in the region for positioning new buildings are identified a priori (i.e., circles); (**B**) centroids of zones to be created are placed randomly in the region (i.e., squares); (**C**) each node in the region is assigned to a centroid based on distance proximity (each node is numbered and assigned to a numbered centroid; matching displayed through the shading in the figure); (**D**) the location of the centroid is recalculated and new nodes are assigned/removed in response; (**E**) the final clusters are created.

Let *A* be the coordinate of the area nodes and let *Q* denote the centroid of the zones to be created in the region. The underlying urban space should already contain the available spaces for region development, referred to as the nodes (see Figure 3A). The algorithm starts by placing an input number of zone centroids in the urban space at random, as demonstrated in Figure 3B. The aim is to then group the nodes into clusters, forming the discrete zones of the region. The algorithm iterates through all nodes present in the region, finding the nearest centroid to each node, according to Equation (1).

$$Q\_j = \arg\min\_j D(A\_{i\prime}Q\_j) \tag{1}$$

where *D*(*Ai*, *Qj*) is a distance function.

Once all nodes have been iterated through, a centroid is allocated to one or more nodes, as demonstrated via the hatching displayed in Figure 3C. The algorithm then recalculates the position of each centroid based on the following equation, Equation (2):

$$Q\_j = \frac{1}{n\_j} \sum\_{i \in V\_j} A\_i \qquad \forall j \in \Pi \tag{2}$$

where set *Vj* is the set of all area nodes allocated to the centroid *j* ∈ Π in the previous step.

Equation (2) considers the average sum of all area node coordinates clustered around a specific zone centroid as the determining attribute of the updated centroid position.

#### 4.1.2. Assignment Model

Once the zone clusters are formed, the next step involves developing an assignment model that allocates each zone to a specific land use in accordance with a given set of criteria. As an example, the criteria can be based on distance to existing roads, soil surface type of each zone cluster, distance to cities/towns nearby, etc. In this study, the assignment model developed is represented via a binary integer programming formulation, where an objective function based on a defined set of criteria is utilised. One of the common criteria used in zoning is the travel distance between zones. As a result, the objective function in this study is formulated to minimise the travel between the different land-use patterns, based on predications of travel of people within the region. Let Π and Λ denote the set of land-use and zonal clusters available, let *akl* denote the people that are expected to travel between land use *k* and *l* (e.g., between commercial and residential zones) and let *bvo* denote the distance of travel between zone cluster *v* and *o*. The integer variable *wkv* is defined to equal to 1 if land use *k* Π is assigned to zone cluster *v* Λ, and 0 otherwise. The objective function to be minimised (total distance of travel between land uses allocated to zones), is given as Equation (3):

$$\sum\_{w, \rho \in \Pi} \sum\_{k, l \in \Lambda} a\_{kl} b\_{\overline{z}w} w\_{k\overline{z}} w\_{l\nu} + \sum\_{v \in \Pi} \sum\_{k \in \Lambda} g\_{k\overline{v}} w\_{k\overline{v}} \tag{3}$$

Essential constraints that are defined are assignment constraints, which specify that each land use that planners intend to position in a given region are assigned to a particular land zone, as given by Equation (4):

$$\sum\_{V \in \Pi} w\_{kv} = 1 \qquad \forall k \in \Lambda \tag{4}$$

The domain of the binary variables is defined in Equation (5):

$$w\_{kv} \in \{0, 1\} \qquad \forall k \in \Lambda \; \forall v \in \Pi \tag{5}$$

Additional sets of constraints that specify other requirements, such as distances that are required between land uses and required connections to existing roads, etc., can also be formulated. It is also important to note that other objective functions that target other criteria can be formulated for assignment of land-use areas to zonal clusters identified. The distance criterion was adopted in this study due to its high relevance in zone planning in urban regions.

#### *4.2. Building Location and Infrastructure Model*

Once the zonal configuration and land use specified for each zone is obtained, the next step involves formulating a model to (i) locate buildings in the region and (ii) determine the investment expansions required for existing infrastructure in response. A bilevel model [42] is proposed which accounts for the decision of two key decision-makers in a smart city setting, namely the urban planners and the transportation network users. The decision space of urban planners is modelled through optimising decisions related to the sustainability of the locations chosen for the buildings within the urban region, along with optimising the infrastructure investment decisions. The decision space of the network users is modelled through optimising their choice of links within the network in response to congestion created and demand generated when an urban planner places new buildings. Since the transport users respond to changes induced by decisions made by urban planners, the model proposed is developed based on a two-level hierarchical system, where the upper level represents the optimisation of the urban planners' decision, while the lower level models the behaviour of users in response to decisions made by urban planners.

The upper level of the proposed model is described next.

#### 4.2.1. Upper Level

The main decision variables in the upper level are: (i) the location decision, represented by the binary variable *z f s* and which equals 1 if building *f* is placed in location *s*, and 0 otherwise; (ii) the binary variable *yij*, which specifies whether link (*i*, *j*) is constructed or not; and (iii) the continuous variable *φij*, which indicates whether an existing link of the network is expanded or not.

#### Upper-Level Objective Functions

The upper-level model involves the formulation of three objective functions; each function targets one specific measure of sustainability. The first equation modelled is a proxy for the social pillar of sustainability (Equation (6)); it minimises the total noise pollution experienced in each zone of the smart city. Noise is generated by the buildings to be positioned in the region, as measured by the parameter *Mrs*.

$$\min\_{z} \sum\_{t \in T} \sum\_{f \in F\_l} \sum\_{s \in P\_l} \sum\_{r \in P} z\_{fs} M\_{rs} \tag{6}$$

Equation (7) targets the economic aspect of sustainability, where the cost of constructing buildings in the zones of the urban region, *Cs* , is minimised.

$$\min\_{\overline{z}} \sum\_{t \in T} \sum\_{f \in F\_l} \sum\_{s \in P\_l} z\_{fs} \overline{C\_s} \tag{7}$$

The final objective function, Equation (8), considers the minimisation of the total carbon emissions from users on the traffic network. Equation (8) accounts for the emissions from different transportation modes, *εm*, which is multiplied by (i) the distance of the links of the network, *dij*; (ii) the flow on the links, *xij*; and (iii) the time of travel on the links of the network, which considers the congestion impacts on the roads, as given by Equations (9) and (10).

$$\min\_{\mathbf{x}} \sum\_{(i,j)\in L^R} \sum\_{m\in \Gamma} \varepsilon\_{m} d\_{ij} \mathbf{x}\_{ij} t\_{ij} \left( \mathbf{x}\_{ij} \right) \tag{8}$$

$$t\_{\vec{ij}}(\mathbf{x}\_{\vec{ij}}) = T\_{\vec{ij}}^{0} \left( 1 + 0.15 \left( \frac{\mathbf{x}\_{\vec{ij}}}{k\_{\vec{ij}}^{0} + \phi\_{\vec{ij}}} \right)^{4} \right) + \left( 1 - y\_{\vec{ij}} \right) M \qquad \forall (i, j) \in L^{N} \tag{9}$$

$$t\_{ij}(\mathbf{x}\_{ij}) = T\_{ij}^{0} \left( 1 + 0.15 \left( \frac{\mathbf{x}\_{ij}}{k\_{ij}^{0} + \phi\_{ij}} \right)^{4} \right) \qquad \forall (i, j) \in L^{R} \backslash L^{N} \tag{10}$$

where *T*<sup>0</sup> *ij* denotes the free flow travel time, while *<sup>k</sup>*<sup>0</sup> *ij* and *φ ij* denote the existing capacity and the upgraded capacity of link (*i*, *j*), respectively.

In particular, Equations (9) and (10) represent the BPR link cost function developed by the Bureau of Public Roads (BPR) [45], which accounts for congestion. Equations (9) and (10) encompass the decisions related to the expansion of the network in order to determine the impacts on congestion levels in the network. Specifically, Equation (9) considers the link addition decisions, *yij*, while Equation (10) applies for all other link types that fall into *LR*, apart from the new links *LN*.

#### 4.2.1.0. Upper-Level Constraints

A number of constraints are defined in the upper-level model to delineate part of the decision space of the urban planners. In particular, Equation (11) specifies that each building is to be positioned in a node within the region.

$$\sum\_{p \in P\_t} z\_{fp} = 1 \qquad \forall t \in T\_\prime \forall f \in F\_t \tag{11}$$

Equation (12) indicates that each node should host at most a single building:

$$\sum\_{f \in F\_t} \sum\_{t \in T} z\_{fp} \le 1 \qquad \forall p \in P \tag{12}$$

Equation (13) is a budget constraint to ensure that investment decisions related to network expansion are kept under control.

$$\sum\_{(i,j)\in L^E} c\_{ij}\phi\_{ij} + \sum\_{(i,j)\in L^N} c\_{ij}y\_{ij} \le B \tag{13}$$

The domain of the upper-level variables is defined by Equations (14)–(17).

$$z\_{fp} \in \{0, 1\} \qquad \forall p \in P, \forall f \in F \tag{14}$$

$$0 \le \phi\_{\vec{i}\vec{j}} \le k\_{\vec{i}\vec{j}}^{0} \qquad \forall (i, j) \in L^{E} \tag{15}$$

$$
\phi\_{i\bar{j}} = 0 \qquad \forall (i, j) \in L^N \tag{16}
$$

$$y\_{i\uparrow} \in \{0, 1\} \qquad \forall (i, j) \in L^N \tag{17}$$

#### 4.2.2. Lower Level

People within the smart city will attempt to reduce their individual travel times when travelling on the transportation network. These decisions will highly depend on the changes induced by decisions made by urban planners, in terms of both the location of new buildings and network expansion decisions related to infrastructure investment.

### Lower-Level Objective Functions

Since the transport network users will attempt to minimise their individual travel times, this sort of selfish behaviour of users can be modelled via a user equilibrium (UE) traffic assignment model [46], such as Equation (18).

$$\min\_{\mathbf{x}} \quad \sum\_{(i,j)\in L^R} \int\_0^{x\_{ij}} t\_{ij}(\omega) d\omega \tag{18}$$

### Lower-Level Constraints

The lower-level constraints focus on the flow variable, to assign traffic to different links within the network. To ensure flow conservation at each node within the network, Equation (19) is defined.

$$\begin{array}{llll}\sum & \mathbf{x}\_{ij}^{\boldsymbol{u}} - \sum & \mathbf{x}\_{ji}^{\boldsymbol{u}} = q\_{\text{lin}} & \forall i \in D \cup P, \forall \boldsymbol{u} \in \mathcal{U}, i \neq \boldsymbol{b}, (\boldsymbol{i}, \boldsymbol{u}) \in \mathcal{W} \quad \text{(19)}\\\ j \in D \cup P: & j \in D \cup L^{V} & \forall \boldsymbol{i}, j \in L^{R} \cup L^{V} \end{array}$$

In order to link the decision variable of the upper level with the decisions made by the lower level, Equations (20) and (21) are defined. In particular, Equation (20) states that flow from the location of the building to the sink node (which accumulates total travel on the network to the particular building type, i.e., single destination) is only possible if that specific building is located on the node from which the link emanates. Equation (21) states that total flow to all destinations on a proposed link within the network, *xij*, is not made possible unless the link is assigned to be constructed.

$$\mathbf{x}\_{pu}^{u} \le z\_{fp}M \quad \forall t \in T, \forall p \in P^t, \forall f \in F^t, \forall u \in D^t \tag{20}$$

$$\text{tr}\_{ij} \le y\_{ij}M' \qquad \forall (i, j) \in L^N \tag{21}$$

Equation (22) is a definitional constraint which specifies that the total flow on a given link is the sum of all flows heading towards all destinations, *u*, on that respective link:

$$\mathbf{x}\_{i\bar{j}} = \sum\_{u \in \mathcal{U}} \mathbf{x}\_{i\bar{j}}^{u} \qquad \forall (i, j) \in L^{\mathbb{R}} \tag{22}$$

The domain of the lower-level decision variable is defined via Equation (23).

$$L\mathbf{x}\_{ij}^{u} \ge 0 \quad \forall (i, j) \in L^{R} \cup L^{V}, \forall u \in \mathcal{U} \tag{23}$$

#### **5. Solution Approach**

The lower-level constraints focus on the flow variable in order to assign traffic to different links within the network. To ensure flow conservation at each node within the network, Equation (19) is defined.

In order to solve the proposed bilevel model above, a procedure which relies on converting the bilevel formulation into a single level model is adopted. A flow chart depicting the major steps undertaken is presented in Figure 4. In particular, the Karush–Kuhn–Tucker (KKT) equivalent conditions are used to reformulate the lower-level model, resulting in a single-level representation. The resulting single level is a mixed integer nonlinear programming (MINLP) model, which is then linearised through implementing a scheme that is based on piecewise approximation of the convex BPR function. Given that multiple objectives are considered at the upper level, a multi-objective optimisation solving approach is required. Lexicographic optimisation [47], which assumes a particular preference order over the criteria included, is adopted. The next section outlines the use of Karush-Kuhn-Tucker (KKT) conditions to reformulate the bilevel model.

**Figure 4.** Solution approach adopted for the bilevel airport location model.

#### *5.1. Equivalent Lower-Level Model*

The UE conditions of the lower-level program can be represented by a set of first-order equivalent constraints, namely the KKT conditions [42]. A dual variable *μiu* is defined for Equation (19). Complementary slackness conditions of KKT, which are equivalent to the UE of the lower level and which require either - *tij* − *μiu* + *μju* = 0 or *x<sup>u</sup> ij* = 0, are enforced by Equations (24) and (25).

$$t\_{ij} - \mu\_{iu} + \mu\_{ju} \ge 0 \qquad \forall (i, j) \in L^R, \mu \in \mathcal{U} \tag{24}$$

$$(t\_{ij} - \mu\_{iu} + \mu\_{ju})x\_{ij}^{u} = 0 \qquad \forall (i, j) \in L^{R}, u \in \mathcal{U} \tag{25}$$

Since Equation (25) involves the multiplication of two variables and is hence nonlinear, the single-level model cannot be solved using a linear solver. To overcome this, an appropriate linearisation scheme to reformulate Equation (25) needs to be applied, as demonstrated in the next section.

#### *5.2. Linearisinng the KKT Conditions*

Let *ωiju* be an auxiliary binary integer variable, which equals 1 if *tij* − *μiu* + *μju* = 0, and 0 otherwise. The complementary slackness condition, Equation (25), is replaced with the following set of constraints, Equations (26)–(28), resulting in the linearisation of KKT conditions:

$$L\mathfrak{x}\_{ij}^{\mu} \le \omega\_{ij\mu} O \qquad \forall (i, j) \in L^R, \mu \in \mathcal{U} \tag{26}$$

$$
\mu\_{\rm ij} - \mu\_{\rm in} + \mu\_{\rm ju} \le \left(1 - \omega\_{\rm iju}\right) \mathcal{O}' \qquad \forall \left(i, j\right) \in L^R, \mu \in \mathcal{U} \tag{27}
$$

$$
\omega\_{ij\mu} \in \{0, 1\} \qquad \forall (i, j) \in L^R, \mu \in \mathcal{U} \tag{28}
$$

where *O* and *O* are large positive constants.

#### *5.3. Linearising the BPR Function*

A chain of linked special ordered sets (SOS) conditions is implemented for linearising the BPR functions in Equations (9) and (10) [48]. The principle idea behind the SOS linearisation scheme adopted is shown in Figure 5. The domains of *xij* and *φij* (i.e., the two continuous variables in the BPR function) are partitioned into *h* ∈ *H* and *e* ∈ *E* regions, respectively, based on a grid point structure. With each grid point, a continuous variable, namely *ψijhe*, is associated. A necessary condition is imposed on *ψijhe*, which states that no more than four adjacent grid points can be nonzero. The flow and link capacity variables can then be represented by Equations (29) and (30), respectively:

$$\mathbf{x}\_{i\bar{j}} = \sum\_{c \in E} \sum\_{h \in H} \overline{\mathbf{x}}\_{i\bar{j}h} \boldsymbol{\upmu}\_{i\bar{j}h\bar{e}} \qquad \forall (i, j) \in L^{\bar{R}} \tag{29}$$

$$\Phi\_{ij} = \sum\_{c \in E} \sum\_{h \in H} \overline{\Phi}\_{ijk} \psi\_{ijhc} \qquad \forall (i, j) \in L^R \tag{30}$$

where *xijh* and *φije* are predefined, fixed values of flow and capacity, respectively, used in the piecewise linearisation of the BPR function.

The BPR function is then approximated by the linear formulation through Equations (31) and (32):

$$t\_{ij} = \sum\_{c \in E} \sum\_{h \in H} T\_0 \left( 1 + 0.15 \left( \frac{\overline{\mathbf{x}}\_{ijh}}{k\_{ij}^0 + \overline{\Phi}\_{ijc}} \right)^4 \right) \Psi\_{ijhc} + \left( 1 - y\_{ij} \right) M \qquad \forall \left( i, j \right) \in L^N \tag{31}$$

$$t\_{ij} = \sum\_{c \in E} \sum\_{h \in H} T\_0 \left( 1 + 0.15 \left( \frac{\overline{\mathbf{x}}\_{ij\overline{h}}}{k\_{ij}^0 + \overline{\Phi}\_{ijc}} \right)^4 \right) \psi\_{ijhc} + (1 - y\_{ij})M \qquad \forall (i, j) \in L^R \backslash L^N \tag{32}$$

The conditions imposed on *ψijhe* are given by Equations (33)–(36):

$$\sum\_{c \in E} \sum\_{h \in H} \psi\_{ijhc} = 1 \qquad \forall (i, j) \in L^R \tag{33}$$

$$\xi\_{ij\hbar} = \sum\_{c \in E} \psi\_{ij\hbar c} \qquad \forall (i, j) \in L^R, \forall \hbar \in H \tag{34}$$

$$
\psi\_{\rm ije} = \sum\_{h \in h} \psi\_{\rm ijhe} \qquad \forall (i, j) \in L^R, \forall c \in E \tag{35}
$$

$$L^{\mathsf{T}}\xi\_{\mathsf{i}\mathsf{j}h\mathsf{\prime}}\eta\_{\mathsf{i}\mathsf{j}c}\in\mathrm{SOS2}\qquad\forall(\mathsf{i},\mathsf{j})\in L^{R},\forall h\in H\_{\mathsf{i}},\forall c\in E\tag{36}$$

Equation (33) is the usual convex combination requirement in piecewise linear approximation. Two auxiliary continuous variables, *ξijh* and *ηije*, are defined and these are embedded within Equations (34)–(36), so that at most, four adjacent variables of *ψijhe* can be nonzero. The combination of Equations (34) and (36) specifies that two adjacent *ψijhe* at most in the *h* ∈ *H* direction can be nonzero, while Equations (35) and (36) state that at most two adjacent *ψijhe* at most in the ∀*e* ∈ *E* can be nonzero. In particular, Equation (36) states that the variables, *ξijh*, *ηije*, are of a special ordered set (SOS) of Type 2 (i.e., SOS2), where a maximum of two of the latter variables that are adjacent can be nonzero. This becomes obvious from Figure 5, since for the grid structure shown and in accordance with the latter equations enforced, not more than two adjacent variables of *ξijh* and *ηije* can be nonzero in the *x* and *y* directions, respectively. The SOS2 conditions are specified as follows in Equations (37)–(40):

$$
\mathbb{Z}\_{ijh} \le \mathbb{Z}\_{ijh-1} + \mathbb{Z}\_{ijh:h \in \overline{H}} \qquad \forall (i, j) \in L^R, \forall h \in H \tag{37}
$$

$$\sum\_{h \in \overline{H}} \mathbb{Z}\_{ijh} = 1 \qquad \forall (i, j) \in L^R \tag{38}$$

$$
\gamma\_{ijc} \le \gamma\_{ijc-1} + \gamma\_{ijc.c \in \overline{E}} \qquad \forall (i, j) \in L^R, \forall c \in E,\tag{39}
$$

$$\sum\_{\mathbf{c}\in\overline{\mathbf{c}}} \gamma\_{\mathbf{i}\mathbf{j}\mathbf{c}} = 1 \qquad \forall (\mathbf{i}, \mathbf{j}) \in L^{\mathbb{R}} \tag{40}$$

The domain of the variables used to mimic SOS2 is given as follows by Equations (41)–(43):

$$\mathcal{L}\_{ijh} \in \{0, 1\} \qquad \forall (i, j) \in L^R, \forall h \in H \tag{41}$$

$$
\gamma\_{ij\varepsilon} \in \{0, 1\} \qquad \forall (i, j) \in L^R, \forall \varepsilon \in E \tag{42}
$$

$$
\psi\_{ijhc} \qquad \forall (i, j) \in L^R, \forall h \in H, \forall c \in E \tag{43}
$$

**Figure 5.** Grids defined for piecewise linearisation of the BPR function.

#### *5.4. Linearing Carbon Emissions Objection Function*

To linearise the carbon emissions objective function, Equation (8) is replaced by the equivalent Equation (44):

$$\min\_{\pi} \sum\_{i \in P} \sum\_{u \in lI} \sum\_{m \in \Gamma} \varepsilon\_{m} d\_{iu} \pi\_{iu} q\_{iu} \tag{44}$$

where *πiu* highlights the shortest travel time between origin *i* and destination *u*.

#### *5.5. Lexicographic Optimisation*

Given that the bilevel model proposed for the urban design of smart cities contains multiple objectives that need to be satisfied, no single solution will optimise all criteria at once. As a result, the concept of optimality adopted in single-objective optimisation is replaced with the concept of Pareto optimality.

A solution *z*∗ of a multi-objective optimisation problem is said to be Pareto optimal if there is no other feasible solution *z* such that *f<sup>θ</sup>* (*z*) ≤ *f<sup>θ</sup>* (*z*∗) ∀*θ* ∈ Θ and *fρ*(*z*) ≤ *fρ*(*z*∗) for at least one index *ρ* ∈ Θ, *θ* = *ρ*, where Θ is the set of objective functions solved in the multi-objective problem.

Lexicographic optimisation involves assigning a preference order over all objective functions considered and solving the problem over a number of stages [49]. In this paper, the lexicographic optimisation approach is adopted, given that it is likely that urban designers have a preference order defined over certain criteria when structuring a region. The algorithm developed is displayed in Algorithm 1.

**Algorithm 1** Lexicographic Optimisation

**Input: Criteria preference** 1 <sup>∗</sup> = 1() **for** = 2, … , Θ <sup>∗</sup> = { (): () ≤ ∗ ∀ = 1, … , − 1} **end for Lexicographic minimiser:** ∗ <sup>∗</sup> {: () ≤ ∀ = 1, … , }

The notation adopted for the lexicographic optimisation process is given by the term *lex* min[*Bv*, *Bw*], which indicates that the model is first solved by minimising the highest ranked objective, *Bv*. Once an optimal solution is yielded, the model is re-solved by adopting *Bw* as the objective function and by including the constraint *Bv* ≤ *B*<sup>∗</sup> *<sup>v</sup>* in the model, where *B*<sup>∗</sup> *<sup>v</sup>* is the optimal solution of *Bv* obtained at the initial stage. The final solution is that attained once all |Θ| − 1 objective functions have been included as constraints, where Θ is the set of all objective functions involved in the model.

#### **6. Computational Results**

In this section, the computational experiments utilised to demonstrate the applicability of the proposed optimisation framework are explained. In the first set of experiments, labelled Scenario 1, the framework is tested on a realistic example of a region being developed into a smart city. The structure of the city is displayed in Figure 6A. Figure 6B displays the available locations for positioning different buildings in the region. The type of buildings considered in the example include schools, hospitals, residential dwellings, offices, bus stops and factories. In the second set of experiments, multiple instances of network structures are generated in order to examine the performance of the proposed model. For both sets of experiments, the proposed mathematical optimisation models are coded in AMPL [50]; Python [51] is used as the programming language to

generate instances in Scenario 2. The model is run on a personal computer with Intel Core i7, 2.2 GHz CPU and 8 GB RAM. CPLEX 12.7 is adopted as the linear solver [52].

**Figure 6.** (**A**) Region examined in the case example; (**B**) available locations for placing buildings in the region.
