Redrawing boundaries for balance: A population-based regional division model

Abstract

Balancing the span of management across administrative divisions is a persistent challenge in urban and regional spatial governance. This study adapts a spatial optimization model originally developed for electoral districting to partition a given region into contiguous and compact divisions while satisfying explicit balance constraints for a chosen target variable, such as population. The model combines an enumeration–recursion search strategy to generate all feasible divisions with a binary programming problem that minimizes deviations from the target balance. Using population as the illustrative variable, we apply the model to redraw prefecture, urban district, and subdistrict boundaries in three respective cases in China—Anhui Province, the urban area of Chengdu, and Shijingshan District, Beijing—and compare the optimization results with the status quo. The results show substantial reductions in population disparity between divisions relative to existing administrative boundaries, indicating a more balanced distribution of population, and therefore a more equal span of management. The model can be extended to various spatial governance tasks, including school districting, police districting, and infrastructure catchment area planning.

Keywords

spatial division span of management administrative boundaries population balance

Introduction

The division of space into administrative units defines the basic structure of territorial governance. It determines how space is partitioned into manageable units for administration, planning, and resource allocation (Kiessling and Pütz, 2020). These spatial units frame how governments—as jurisdictions—act, and how residents interact with the state (Grossman and Lewis, 2014). The principles behind such division are often historically inherited and geographically constrained (Skinner, 1977). However, as transportation technologies reduce spatial frictions and significantly improve mobility, the question of how space should be divided—on what basis, and for what purpose—requires renewed attention (Batty, 2008; Lin, 2001). As a result, spatial division must now be examined through new methodological lenses. In particular, the effectiveness of spatial governance should be treated as an explicit objective of division design.

One key dimension of effectiveness of administrative division is the span of management—the scale and complexity of what each unit must manage (Gulick, 1937; Simon, 1946). The factors shaping span of management in public administration may involve land area, population, infrastructure, or public service provision (Andrews et al., 2005; Ostrom et al., 1961). When these factors vary widely across administrative divisions at the same level, governance becomes uneven: some jurisdictions face temporary or chronic overload; others operate below capacity. Although the institutional hierarchical structure differs across countries, the challenge of uneven span of management between administrative divisions is broadly echoed.

In federal systems such as the United States, administrative units with vastly different sizes and responsibilities often hold equal institutional roles. California and Alaska, for example, have equal Senate representation despite sharp differences in territorial extent, population, and governance demands (Berkowitz and Krause, 2020). Similar issues appear in India, where districts within the same state range from under one million to over seven million residents, despite holding equal administrative status (Tiwari et al., 2020). Similar issues also emerge in the Russian context: municipal units within the same federal subject often face divergent governance demands despite holding equal administrative status (Medvednikova, 2024). In China, cities of the same administrative tier vary dramatically in both land area and population, creating divergent governance pressure (Zhang et al., 2025). For example, Suzhou and Hegang, both prefecture-level cities, had in 2023 about a 13-fold difference in population and a 65-fold difference in economy. These cases reflect a broader pattern: spatial division often fails to align governance responsibilities with actual demand.

This misalignment motivates the research question of this study: how can administrative divisions be structured such that governance responsibilities are more evenly aligned with actual management demands across territorial units at the same institutional level? Addressing this question requires moving beyond descriptive comparisons of unit size toward a formal framework for evaluating and redesigning spatial divisions under explicit balance constraints. In this study, we propose a computational approach to administrative regionalization that reframes a classical electoral districting model for spatial governance analysis. While the original model proposed by Garfinkel and Nemhauser (1970) was developed for electoral districting—where equal population is a normative requirement tied to political representativeness and concerns such as gerrymandering—the logic of administrative division is fundamentally different. Here, population balance does not serve as a proxy for representational equality, but rather reflects differences in governance demands, service provision pressure, and administrative workload across territorial units at the same institutional level. Reinterpreted in this way, population balance functions as a managerial constraint rather than a representative one. Building on this reframing, we operationalize the model using contemporary optimization tools and apply it across multiple administrative scales, demonstrating how a theoretically simple spatial optimization framework can be repurposed to evaluate and redesign administrative boundaries in support of more balanced spatial governance.

The rest of the paper is organized as follows. The next section reviews relevant literature on spatial division methods and the concept of span of management, positioning our study within these domains. We then present the spatial division model, which integrates contiguity, compactness, and a balance constraint for a chosen target variable. This is followed by three case studies that apply the model to redraw prefecture, urban district, and subdistrict boundaries in China, using population as the illustrative balance variable. The article concludes with a conclusion and discussion section.

Literature review

Designing how space is divided into administrative or functional divisions is a core problem in territorial governance (Faludi, 2012). Well-designed spatial divisions in public administration can balance government workloads, improve service delivery, and enhance administrative efficiency, while poorly designed ones can entrench inequalities and inefficiencies (Chao et al., 2023). Scholarship on spatial division spans sociology (Donnelly and Gamsu, 2022), geography (Wang and Feng, 2021), planning (Knaap and Hopkins, 2001), public administration (Ostrom et al., 1961), and operations research (Daskin, 2011), with different traditions emphasizing physical geography, governance structures, or optimization techniques. This review focuses on two strands most relevant to our study: (1) research on spatial division methods, particularly optimization-based approaches for partitioning a region into contiguous and compact divisions under explicit constraints; and (2) research on the span of management in public administration, which examines the equal distribution of administrative responsibilities.

Spatial division methods

The primary objective of spatial division is to address heterogeneity in regional environmental conditions and conflicts in spatial utilization (Chen and Cai, 2022), thereby enabling a more balanced distribution of resources and environmental quality across subregions (McHarg, 1969). As a tool for spatial governance, spatial division is widely used in territorial spatial planning (Fan et al., 2019), administrative restructuring (Zhang et al., 2022), and the allocation of public resources such as health care and education (Dai et al., 2025; Gu et al., 2010).

Existing studies on spatial division can broadly be divided into three strands. The first adopts a managerial and qualitative perspective, evaluating the rationality and developmental implications of division schemes (Menzori et al., 2021). While such analyses are valuable, they often cannot fully capture the spatial variability inherent in regional governance conflicts (Bao et al., 2021). The second strand focuses on optimization-based approaches, where spatial operations research—an intersection of operations research (OR) with geographic information science (GIS), spatial analysis, and related fields—provides an effective perspective for delineating divisions (Alagador and Cerdeira, 2022). However, solving complex division problems under multiple realistic constraints remains challenging. The third examines specific applications at different spatial scales, such as territorial spatial planning with ecological restoration and carbon neutrality as objectives (Ran et al., 2024). Across these strands, the equality dimension of spatial division is often overlooked in favor of efficiency or environmental goals.

Span of management

Equality in spatial division is closely related to the design of span of management across divisions. The concept of span of management originates in public administration (Gulick, 1937), often discussed alongside hierarchical levels—the number of vertical tiers in an authority structure, ranging from decision-making to implementation. Under a fixed organizational scale, hierarchical levels and the span of management are inversely related: more levels imply narrower spans, fewer levels imply wider spans (Ouchi and Dowling, 1974).

A balanced span of management can have important implications for national security, administration, and regional development (Gulick, 1937; Keating, 2013). From an administrative perspective, an appropriate span is a prerequisite for effective governance and an important determinant of administrative efficiency (van Meerkerk and Edelenbos, 2018). Both excessively large spans, which strain administrative resources, and overly small spans, which cause redundancy, have been found to reduce administrative effectiveness (Gulick, 1937; Ouchi and Dowling, 1974; Urwick, 1956). From an economic perspective, maintaining a moderate span can promote regional economic development (Wang and Feng, 2021), and empirical evidence shows that appropriately sized administrative divisions, such as those achieved through city–county mergers, can enhance integration and governance capacity (Ma et al., 2024). These findings support the practical importance of designing spatial divisions with a balanced span of management.

Research gap

Although spatial optimization models have been extensively developed in the operations research literature, and the span of management has been conceptually and empirically examined in public administration studies, the two streams have rarely intersected. The equal distribution of administrative workload—operationalized here as a balanced span of management—has not been systematically formulated as a spatial optimization problem. Existing spatial division models, including those for political districting, have seldom been applied to territorial governance problems in which the balance in span of management is the primary objective.

This study addresses that gap by adapting the integer programming framework for electoral districting proposed by Garfinkel and Nemhauser (1970) to the context of equalization in span of management. The target variable in this framework is generic; in this article we use population as a representative example.

Method

As discussed above, we adapt the optimization framework originally proposed by Garfinkel and Nemhauser (1970) very specifically for electoral districting in the U.S. to a broader set of spatial division settings. The model partitions a given region into contiguous and compact divisions while optimizing an explicit balance objective for a chosen target variable.

In this article, we illustrate the approach using population as the balance variable in three case studies of provincial, prefecture, and county-level boundary delineation in China. We would like to remark here that the use of population is intended as a baseline specification. In practice, the target variable can be adjusted to reflect policy-relevant factors. For example, the raw population P_i may be replaced with a weighted population measure that incorporates local service levels (e.g., adjusted by public transport accessibility or Point-of-Interest density). This does not alter the model structure, but modifies the balancing criterion within the same formulation. The solution of the model integrates an enumeration–recursion search to generate feasible divisions, and a binary programming problem that minimizes deviations from the target balance. In the following, we first introduce the model’s structure and constraints, and then the solution procedure.

The overall idea of the model is to operationalize three widely recognized constraints in spatial division: balance for a chosen target variable (Ricca et al., 2013), contiguity (Shirabe, 2005)—such that each division is not split in many parts, and compactness (Murray, 2025)—such that each division is relatively “circular” rather than tenuous. Before we start to formulate the model, we list the parameters and variables used for the model in Table 1.

Table 1.

Notations of variables in the model.

Formulation variables	Definition
N	Number of basic units
M	Number of divisions to be divided
S	Number of feasible divisions
X_j	Binary variable: 1 if feasible division j is in the solution, 0 otherwise
Y_ij	Binary variable: 1 if basic unit i belongs to feasible division j, 0 otherwise
α	Population slack parameter
β	Compactness control parameter
P_i	Population of basic unit i
Q_j	Total population of feasible division j
$\bar{Q}$	Average population per division
D_j	Population deviation of division j from $\bar{Q}$
d_ik	Distance between basic unit i and basic unit k
E	Maximum distance between basic units
C_j	Compactness score of division j
Solution variables	Definition
V	Set of basic units selected for expansion to generate a feasible division
Q_v	Total population of basic units in V
FV	Set of feasible divisions

Model formulation

Consider a general case, as illustrated in Figure 1: a spatial region is composed of N basic units (the small boxes), each associated with a measurable attribute—such as population P_i. The task is to aggregate these units into divisions (such as the black lines enclosed parts), thereby partitioning the region. The goal is to ensure that the total attribute value of each division—calculated as the sum of the values of its constituent basic units—is as balanced as possible.

Figure 1.

Illustration of the model concept.

More formally, if we are going to divide a region into M divisions, we define the objective function to be to minimize the maximum population deviation of a division from the average:

\min \max_{j = 1, \dots, S} D_{j} X_{j},

(1)

where $D_{j} = | Q_{j} - \bar{Q} | / (α \bar{Q})$ is defined as the relative population deviation in a potential division $j$ . Initially, more than M feasible divisions are generated, and we denote the total number of such feasible divisions to be S. A binary variable X_j is introduced to indicate whether division j is selected into the final optimal solution. Among the S feasible candidates, exactly M will be selected to form the optimal partitioning, which will be guaranteed in the constraints.

There are eight constraints in the optimization model, organized into two groups, as follows. The first four define the conditions that each feasible division must satisfy. The latter four ensure that the final selection of feasible divisions constitutes a legitimate partition of the entire region—all basic units are covered exactly once, and with no overlaps, or technically, mutually exclusive and collectively exhaustive.

| Q_{j} - \bar{Q} | \leq α \bar{Q}

(2)

b_{i k} = 1, \forall i \exists k in the same division

(3)

d_{i k} \leq E, \forall i, k in the same division

(4)

C_{j} \leq β, j = 1, \dots, S

(5)

X_{j} = 0, 1, j = 1, \dots, S

(6)

Y_{i j} = 0, 1, i = 1, \dots, N, j = 1, \dots, S

(7)

\sum_{j = 1}^{S} X_{j} = M, j = 1, \dots, S

(8)

\sum_{j = 1}^{S} Y_{i j} X_{j} = 1, i = 1, \dots, N, j = 1, \dots, S

(9)

The first group of Constraints (2–5) ensures that each feasible division meets three conditions: balanced population, spatial contiguity, and geographical compactness. First, Constraint (2) preliminarily controls population balance. The population of each feasible division Q_j must fall within a specified range around the average population $\bar{Q} = \frac{\sum_{i = 1}^{N} P_{i}}{M} .$ A slack parameter $α (0 < α < 1)$ bounds the tolerance range. A smaller α imposes stricter limits, reducing the number of feasible divisions by excluding those that are too large or too small, thereby improving computational efficiency. Second, Constraint (3) ensures contiguity. A binary variable b_ik is used to require that each basic unit in a feasible division must be adjacent to at least one other unit within the same division. In other words, each feasible division must be a single piece (see details about how this constraint is implemented in the solution section below). Third, Constraints (4) and (5) control the compactness of feasible divisions. This is addressed in two steps: (i) a maximum intra-division distance E limits the distance between any two basic units in the same division; and (ii) a compactness index $C_{j} = \frac{\max {(d_{i k} \times Y_{i j} \times Y_{k j})}^{2}}{A r e a (j)}$ is used to maintain compactness. A smaller C_j indicates a more circular, compact division. A parameter β > 0 sets the upper bound for acceptable compactness.

Constraints (6–9) ensure the validity of the final solution to be a legitimate partitioning of the region: every basic unit is assigned to exactly one division, and no overlaps between divisions. This is operationalized using two sets of binary decision variables: X_j indicates whether feasible division j is selected into the final solution, and Y_ij indicates whether basic unit i belongs to division j. Constraint (8) enforces that exactly M divisions are selected (i.e., exactly M of the X_j’s equal 1), thereby achieving an M-partition of the region. Constraint (9) ensures that each basic unit is assigned to one and only one of the selected feasible divisions, i.e., among all divisions with X_j = 1, each basic unit i is associated with exactly one Y_ij = 1.

It is worth noting that additional constraints or considerations can be incorporated into this optimization model. For instance, if preserving continuity with existing administrative boundaries is a policy objective, the model can be extended by introducing a penalty term $λ Σ_{i, k = 1 .. N} q_{i k}$ into the objective function, in which $λ$ is a weighting parameter that controls the trade-off between functional optimality and boundary preservation, and $q_{i k} = 1$ if basic units i and k belong to the same existing administrative division but are assigned to different divisions in the proposed solution. In the results presented in this paper, we do not impose this penalty term, as the degree of continuity with existing boundaries is treated as context-dependent rather than a universal modeling requirement.

Similarly, other considerations, such as historical or cultural similarity among basic units, can also be incorporated by introducing additional weights or constraints that favor the co-assignment of units sharing such attributes. Whether such similarity should be prioritized is likewise context-dependent. In the Chinese administrative context, for example, there have also been policy orientations that intentionally adopt interlocking spatial arrangements to weaken territorial closure and encourage cross-boundary interaction (Han, 2025). This illustrates that administrative boundary design may pursue different and sometimes competing objectives across governance settings. Accordingly, cultural similarity is not imposed as a default constraint in the baseline model, but can be incorporated when it constitutes a primary policy objective and relevant data are available.

Geographic considerations can also be incorporated into the model as additional constraints or extensions. Factors such as terrain or accessibility can be reflected through modified adjacency definitions (e.g., based on transportation costs). These extensions do not alter the core structure of the model, but refine its application in specific contexts.

To provide a benchmark for evaluating the proposed optimization model, we also construct a population-weighted Voronoi partition for each study area. Voronoi-based partitioning is a commonly used and transparent distance-based method in geographical analysis (Fortune, 1986) and serves here as a benchmark. Population weighting is applied in the Voronoi model to reflect population-balance considerations; however, the method does not explicitly optimize or constrain population equality.

Existing administrative divisions are included separately as a status quo benchmark, representing current institutional arrangements rather than an algorithmic regionalization scheme.

Solution

The solution to the optimization problem formulated above begins by fixing the number of divisions M, which could be predetermined based on the existing number of administrative divisions, or a meaningful target (e.g., to reduce the administrative divisions to M to improve administrative efficiency). In this article, M is treated as an exogenously specified scenario parameter rather than an endogenous decision variable. It represents alternative administrative or governance configurations that may be considered by decision-makers, rather than a policy prescription determined by the optimization itself. By fixing M and examining multiple values, the model allows comparison of population-balance outcomes under different hypothetical division structures.

Then, two parameters need to be generated by iteration: the slack parameter of population deviation α, and the compactness bound β. Technically, α is initialized at 0.1 and gradually tightened in increments of 0.01 after a feasible solution is found, continuing until no feasible solution exists. Similarly, β is incremented from 1.5 up to 5. The tuning sequence follows an iterative strategy: (1) fix M, (2) adjust β upward until a solution appears, and then (3) reduce α for refinement. If no solution is found within the allowed β range, limited adjustments to M may be attempted.

The solution process is composed of two stages, in which the first stage is an enumeration–recursion algorithm designed to generate all feasible divisions of a study area under population balance and compactness constraints. The overall workflow of Stage 1 is illustrated in Figure 2.

Figure 2.

Flowchart of Stage 1 of the optimization problem solution.

The solution algorithm operates in two nested loops. In the outer loop, each basic unit is taken in turn as the starting seed, indexed by i, to generate feasible divisions. In each iteration, the index i is placed into the set V, which stores the basic units currently included in the candidate division. The outer loop ensures that every basic unit in the study area serves as a starting point for at least one iteration.

The inner loop iteratively expands V by adding one neighboring basic unit k at a time. Eligible neighbors are those that satisfy both the adjacency and distance constraints defined in Constraints (3–4). Among these, the unit with the smallest population (P^k) is selected first. After each addition, the total population of the division based on $V$ ( $Σ_{k} P_{k}$ for all $k \in V$ ) is evaluated against the lower and upper bounds:

(1 - α) \bar{Q} \leq Σ_{k \in V} P_{k} \leq (1 + α) \bar{Q}

(10)

where α controls the allowed population deviation. If the upper bound is exceeded, the current trial of feasible division is abandoned, and the algorithm backtracks to the previous configuration of V. If the lower bound is not yet reached, the algorithm continues expanding by adding the next smallest eligible neighbor.

When the population bounds are satisfied, the algorithm checks the compactness of the candidate division. If the compactness threshold is met, the current V is recorded as a feasible solution and stored in the set FV (the collection of all feasible divisions found). The inner loop then tests whether V can be further expanded without violating the constraints; if so, the expanded set is also checked for compactness and stored if feasible.

Essentially, the first stage uses a recursive approach to ensure the exhaustion of all possible feasible divisions: once a branch of the search is exhausted—either because no further eligible neighbors remain or because a constraint is violated—the algorithm reverts V to its last viable configuration and explores alternative neighbor choices. This systematic exploration guarantees that all possible divisions meeting the constraints are identified.

The process continues until the outer loop has cycled through all basic units in the study area, at which point the set FV contains the complete list of feasible solutions.

The second stage is given the set of feasible solutions, of size S, to find the M ones within it that meet the optimization problem (1), a well-defined binary optimization problem, which now only needs to satisfy Constraints (6–9). The Gurobi solver (Gurobi Optimization, 2023) is used to solve this simple-form binary programming problem.

Model application

In this article, we select the re-partitioning of administrative divisions in China, using population balancing as a representative case. This choice is based on three considerations. First, China is undergoing administrative division reform, with growing attention to the optimization of territorial governance structures (State Council of the People’s Republic of China, 2018). Second, significant population imbalance exists among administrative divisions at the same level, which leads to disparities in the span of management and administrative efficiency. For example, within Sichuan Province, the largest subdivision, Chengdu (a prefecture-level city) had a population of 20.94 million in 2020, accounting for 25.02% of the provincial total, while Panzhihua (another prefecture-level city) had only 1.21 million in the same year, accounting for just 1.45%. Similar disparities are also found at the county level and the township level. Third, China offers wide availability of reliable population data at multiple administrative levels, providing a data basis for empirical analysis.

Study areas

China’s local governments are organized into four administrative levels: provinces, prefecture-level divisions (primarily prefecture-level cities), county-level divisions (urban districts, counties, and county-level cities), and township-level divisions (subdistricts or jiedao, towns, and townships). In this article, we use cases from the upper three levels in China—provincial, prefectural, and county levels—as the regions to be divided, and select one representative case from each level. Specifically, we choose: (1) Anhui Province for the re-division of prefecture-level cities; (2) the central urban area of Chengdu for the re-division of urban districts; and (3) Shijingshan District¹ in Beijing for the subdivision of subdistricts. The geographic locations and population distributions of the three selected cases are shown in Figure 3.

Figure 3.

Locations and population distributions of the three cases: (lower left) case locations. (a) Anhui. (b) Chengdu Urban Area. (c) Shijingshan District, Beijing.

Anhui Province shows a significant population imbalance among its prefecture-level cities. As shown in Figure 3a (where black lines indicate the existing boundaries of divisions in this case study, as in subsequent figures), population is concentrated in the central to northwestern parts (the population is shown for basic units, same below)—especially in the capital city, Hefei (9.37 million, 15.35% of the provincial total)—while southern cities such as Tongling have much smaller populations (1.31 million, 14% of Hefei’s population, or about 2% of the provincial total).

For the prefecture-level case, Chengdu has 12 municipal districts, where population is unevenly distributed. As shown in Figure 3b, most residents live in the central districts (e.g., Wuhou, Jinniu), while peripheral districts have much smaller populations.

At the county level, Shijingshan District in Beijing consists of nine subdistricts also with imbalanced population distributions. As shown in Figure 3c, Bajiao Subdistrict (104,611) and Lugu Subdistrict (85,716) hold most of the population, while Guangning Subdistrict has only 15,356 residents—just 14.7% of Bajiao’s.

These three study areas are used to demonstrate the proposed method by redistricting their respective administrative divisions—prefecture-level cities, urban districts, and subdistricts—based on smaller basic units—counties for provinces, towns or subdistricts for urban areas, and communities (shequ) for urban districts (which in principle are two levels lower than the region to be divided). All three cases reflect common challenges of uneven population distribution across administrative divisions, leading to governance inefficiencies: densely populated divisions face overstretched services and management burdens, while sparsely populated ones suffer from resource underutilization.

Data sources

As indicated in the method, the data required for applying the model include population data, from the 2020 National Census in China (National Bureau of Statistics, 2021), administrative boundary data, land area, adjacency, and pairwise distance between basic units. Specifically for the provincial-level calculation, we merge adjacent urban districts in the central areas of prefecture-level cities into a single basic unit, since these districts historically originate from the same county-level division. The administrative boundaries are obtained from Tianditu, the National Platform for Common GeoSpatial Information Services in China (Tianditu, 2024). The land area of each unit is calculated using QGIS. Adjacency and distance matrices are calculated based on the administrative boundary data using Python: adjacency is determined by shared boundaries, and distances are computed as great-circle distances between unit centroids. Due to recent adjustments to administrative divisions, there are inconsistencies between census statistical units and current boundaries—especially in the latter two cases—so we adopt the method proposed by Chen et al. (2024) to realign population data with the updated units.

Results

Based on the prepared data, we implemented the optimization model in the three selected study areas. The results are evaluated based on the objective of population balance (i.e., reduced inter-division population variability), measured by the range and standard deviation of population across divisions. As a benchmark, the results are also compared with those of the existing administrative boundaries, which—although not formally included as constraints in the article—represent a practical consideration in redistricting decisions.

We begin by fixing M at the status quo value to isolate the effect of spatial reconfiguration. Under this setting, we compare three configurations: the proposed optimization model (GN model; Garfinkel and Nemhauser, 1970), a population-weighted Voronoi partition (hereafter referred as the VD model), and the existing administrative divisions.

As shown in Figure 4 and Table 2, the GN model consistently achieves substantially improved population balance, with markedly reduced ranges and standard deviations. The VD partition also improves upon the status quo, but its gains are smaller and less consistent than those of the optimization model.

Figure 4.

Comparison of spatial divisions generated by the Garfinkel–Nemhauser optimization model (GN) and a population-weighted Voronoi partition (VD).

Table 2.

Comparison of population distribution metrics across spatial division schemes under the GN model, a population-weighted Voronoi benchmark (VD), and existing administrative divisions (population in thousands).

Study area	Scheme	M	Min	Max	Range	Std. dev.
Anhui Province	Status quo	16	1,311.7	9,369.9	8,058.2	2,332.1
	VD model	16	1,738.0	6,574.8	4,836.8	1,436.8
	GN model	16	3,474.7	5,118.2	1,643.4	394.8
Chengdu Urban Area	Status quo	12	363.6	2,067.8	1,704.2	495.9
	VD model	12	363.6	2,346.8	1,983.2	638.6
	GN model	12	1,224.5	1,302.6	77.9	19.8
Shijingshan District	Status quo	9	15.4	104.6	89.2	25.7
	VD model	9	15.2	100.4	85.2	24.9
	GN model	9	59.4	66.8	7.4	3.3

These results demonstrate that the proposed optimization model can effectively improve population balance. However, the current number of administrative divisions is not necessarily optimal with respect to population balance. To further explore alternative configurations, we apply multiple values of M for each case, considering values both above and below that of the status quo. These values are selected to illustrate the model’s behavior over a reasonable range, rather than to exhaustively enumerate all possible configurations.

Anhui Province

To apply the optimization model to Anhui Province, we tested three values of M—denoted as AH Province 1 (M = 12, α = 0.10, β = 1.5), AH Province 2 (M = 10, α = 0.07, β = 1.5), and AH Province 3 (M = 9, α = 0.06, β = 1.5). The choices of α and β have been explained in the Method section. The division results are shown in Figure 5, with new boundaries of prefecture-level cities indicated by black lines and labeled C1 through Cm. For comparison, the original divisions are shown in color.

Figure 5.

Redistricting results under three scenarios in Anhui.

In the AH Province 1 scenario, many original prefecture boundaries are preserved. For example, in the sparsely populated south, Huangshan, Chizhou, and Tongling are merged into a single unit (C1) without internal division, and most existing prefectures remain largely intact. In contrast, the AH Province 2 scenario results in smaller divisions in the densely populated northern part (e.g., C6–C9), while the sparsely populated southern part (e.g., C1, C2) forms much larger divisions. Under AH3, where the number of divisions is reduced to nine, each new prefecture integrates multiple existing prefectures. For instance, C1 in AH Province 3 scenario consists of county-level units from six different original prefectures, representing the most extensive spatial aggregation among the three scenarios.

Table 3 compares population balance across the three scenarios, using the existing division (denoted as AH Province 0) as a benchmark. All three significantly improve population equality, reducing the population range (maximum minus minimum) from over 8 million to approximately 600 thousand, and lowering the standard deviation from 2,332.1 to below 220. Among them, AH Province 1 achieves the best population balance, with a standard deviation of 199.7 and a range of 610.1, resulting the most even distribution of population across divisions. It also better maintains consistency with existing administrative boundaries, providing a more feasible reference for potential redistricting.

Table 3.

Population distribution metrics for redistricting scenarios in Anhui (in thousands).

Scenario	M	Min	Max	Range	Std. dev.
AH Province 0	16	1,311.7	9,369.9	8,058.2	2,332.1
AH Province 1	12	4,755.3	5,365.4	610.1	199.7
AH Province 2	10	5,845.1	6,498.4	653.3	219.9
AH Province 3	9	6,498.4	7,130.5	632.1	214.7

Chengdu Urban Area

To isolate the effect of the number of divisions, we test alternative values of M both above and below the status quo (12 districts) in the Chengdu Urban Area. Specifically, we consider three scenarios: M = 20, 16, and 11, while keeping other parameters fixed (α = 0.10, β = 2.0). The resulting configurations are shown in Figure 6.

Figure 6.

Redistricting results under three scenarios in the Chengdu Urban Area.

In the Chengdu case, varying M leads to systematic differences in both spatial structure and population balance. Increasing M (e.g., M = 20) produces finer partitions, particularly in the city center, with all existing district boundaries substantially reconfigured. Decreasing M (e.g., M = 11) leads to consolidation, with several peripheral districts merged and a greater degree of continuity with existing administrative units.

As reported in Table 4, all optimized configurations improve population balance relative to the status quo. However, the degree of improvement is not monotonic in M. The intermediate case (M = 16) achieves the best overall balance, with both the smallest population range and standard deviation. This suggests that population balance depends not only on spatial reconfiguration, but also on selecting an appropriate institutional scale. Excessively large or small values of M may reduce this benefit, reflecting a trade-off between subdivision and consolidation.

Table 4.

Population distribution metrics for redistricting scenarios in the Chengdu Urban Area (in thousands).

Scenario	M	Min	Max	Range	Std. dev.
CD Urban Area 0	12	363.6	2,067.8	1,704.2	495.9
CD Urban Area 1	20	721.1	789.4	68.3	21.6
CD Urban Area 2	16	927.8	957.7	29.9	9.2
CD Urban Area 3	11	1,224.6	1,458.0	169.7	69.1

Shijingshan District

For Shijingshan District, we also tested three M’s: SJS District 1 (M = 15, α = 0.10, β = 5.0), SJS District 2 (M = 14, α = 0.05, β = 5.0), and SJS District 3 (M = 13, α = 0.10, β = 1.5). The redistricting results are shown in Figure 7.

Figure 7.

Redistricting results under three scenarios in Shijingshan District.

In the SJS District 1 scenario, the spatial adjustment follows a familiar pattern: sparsely populated areas such as Guangning Subdistrict form new divisions with larger territorial extents, while densely populated areas like Bajiao Subdistrict are split into smaller administrative units. Compared to this, the SJS District 2 scenario generates more different spatial division from the existing ones. In SJS District 3, the resulting boundaries show minimal deviation from existing subdistricts, with population balance achieved largely through internal reconfiguration rather than complete spatial restructuring. For instance, D4 and D5 are both derived from subdividing Pingguoyuan Subdistrict.

As shown in Table 5, all three scenarios in Shijingshan also outperform the existing division (SJS District 0) in terms of population equality. Among them, SJS District 2 provides the most balanced outcome, with a range of just 1.5 thousand and a standard deviation of 500.

Table 5.

Population distribution metrics for redistricting scenarios in Shijingshan District (in thousands).

Scenario	M	Min	Max	Range	Std. dev.
SJS District 0	9	15.4	104.6	89.2	25.7
SJS District 1	15	34.6	40.6	6.0	1.9
SJS District 2	14	39.9	41.4	1.5	0.5
SJS District 3	13	41.8	45.6	3.8	1.5

It is worth noting that, although existing administrative boundaries are not imposed as constraints or a part of the objective function in the model, the resulting spatial divisions exhibit varying degrees of alignment with the current administrative structure. Some scenarios—such as AH1 and SJS3—retain a larger share of existing boundaries, while others involve more substantial spatial reconfiguration. This variation reflects differences in local spatial structure and demand patterns rather than an explicit preference for boundary preservation. As discussed in the Method section, continuity with existing boundaries can be incorporated if desired by introducing a penalty term that controls the extent of deviation between proposed and current divisions; however, this constraint is not imposed in the analysis here, as tolerance for boundary adjustment is treated as context-dependent.

Conclusion and discussion

In this article, we formulated a spatial division model that partitions a region into divisions meeting explicit constraints on contiguity, compactness, and balance for a defined target variable. We applied the model to redraw administrative boundaries at the provincial, prefecture, and county levels in three respective cases in China. In all three cases, the optimized divisions reduced variation in population size between administrative divisions compared with the original boundaries, thereby decreasing disparities in span of management, and presumably improving the balance of workload across divisions.

We would like to note, nonetheless, that population balance represents only one dimension of spatial governance. Administrative divisions are expected to serve multiple functions simultaneously, including public service provision, governance coordination, and territorial management. As a result, optimizing for population balance alone may introduce trade-offs with other objectives, such as equity in service access or administrative capacity, which needs to be fully examined in future research.

Administrative boundaries are also embedded in historical, cultural, and institutional contexts that shape residents’ identities and patterns of interaction. Existing divisions often reflect long-standing historical processes and cultural affinities, and boundary adjustments may disrupt established social relations and local identities. Preserving regions with shared historical or cultural characteristics may thus be an important consideration in some settings, even if doing so constrains the degree of population equalization that can be achieved. Conversely, existing boundaries should not be regarded solely as institutional legacies to be preserved; they may also embody historical compromises or outdated arrangements that are misaligned with contemporary governance demands. Administrative boundary design therefore involves an inherent tension between continuity and adaptation, rather than a simple choice between preservation and optimization.

The empirical cases also illustrate the role of the number of divisions (M) as a governance parameter rather than a decision variable determined by the model. Substantively, M can be interpreted in relation to administrative hierarchy and span of management: larger values of M imply a wider span of management for the higher-level government, while smaller values imply a narrower span. Our optimization model does not determine which M is preferable in an absolute sense. Instead, by fixing M and examining multiple values, the model provides a structured way to evaluate how alternative administrative configurations shape population balance and spatial structure. Decisions regarding the appropriate M ultimately depend on institutional capacity, governance objectives, and political considerations that lie beyond the scope of spatial optimization alone.

Against this background, the proposed model should be understood as an analytical and exploratory tool rather than a prescriptive solution for administrative restructuring. By formalizing population balance as an explicit constraint, the framework enables systematic comparison of alternative spatial division schemes and clarifies the consequences of different design choices. Additional considerations—such as historical continuity, cultural similarity, political feasibility, or administrative requirements—can be incorporated as supplementary constraints or weighting parameters when relevant data and policy priorities warrant their inclusion. There is no universally optimal administrative division, and boundary adjustment remains a context-dependent process.

In the meantime, against the backdrop of increasing mobility, the necessity of balancing populations based solely on residential distribution may be reduced. Although many governance functions still operate through relatively stable territorial units and rely on resident population as a key reference, where population balance remains a relevant objective, alternative formulations based on flow-derived measures are also feasible, depending on the governance context (Jiang et al., 2026).

Footnotes

ORCID iD

Hongmou Zhang

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was partially supported by the Institute of Public Governance, Peking University.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

The data used in this study are publicly available from the National Bureau of Statistics of China.

Notes

Author biographies

Longbin Kou is a PhD student at the School of Government, Peking University. His research interests include public administration and urban geography, with a particular focus on administrative divisions in China.

Lidan Pan is a master’s student at the School of International and Public Affairs, Shanghai Jiao Tong University. She was previously an undergraduate student at the School of Government, Peking University. Her research interests include emergency management, urban public services, and urban governance.

Hongmou Zhang is an Assistant Professor at the School of Government, Peking University.

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