# Optimal Regional Allocation of Future Population and Employment under Urban Boundary and Density Constraints: A Spatial Interaction Modeling Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Historical Overview of Urban Modeling

#### 1.2. Spatial Interaction Modeling: Structure and Variables

_{ij}= kR

_{i}W

_{j}/D

_{ij}

^{α}

_{ij}is the flow from origin i to destination j, R

_{i}and W

_{j}are the measures of the sizes of the origin i and destination j, respectively, D

_{ij}is the distance between them, and α is a positive parameter that represents the distance friction. The R

_{i}and W

_{j}variables are proxies for the abilities of the origin to generate flows and of the destination to attract them. Generalized versions of Equation (1) include several variables that characterize both the origin and destination, and several friction factors. This model has been termed as unconstrained SIM. The estimation of (1), subject to the given total outflows for all the origins and given total inflows for all the destinations, is termed as constrained or entropy SIM. The focus here is on the unconstrained case. These models consider aggregate flow data (e.g., the number of commuters between the origins and destinations). Another interpretation of SIM is related to discrete choice models (e.g., multi-nomial logit models), using disaggregate data at the level of the individual decision maker. Anas [2] argues that the gravity and discrete choice models are two equivalent views of the same problem. For reviews of the theory and applications of SIM, one can refer to the work of Sen and Smith [3], who discuss the theoretical foundations and practical applications of gravity models to commuting, and Nijkamp and Ratajczak [4], who review the relevance of gravitational principles in regional science and spatial economics, and address their application to trade flow analysis.

#### 1.3. Planning Optimization Models and Spatial Interaction Modeling

#### 1.4. Sprawl versus Compact City: Cost Assessment

_{2}emissions still tend to be lower in more compact developments. Stone [46], using data on 45 large U.S. metropolitan areas, shows that sprawling areas are associated with more ozone exceedances than more spatially compact metropolitan regions. Schindler and Caruso [47] develop a theoretical monocentric urban model to analyze the trade-off between traffic-based pollutant emissions and pollution exposure. Solving the model with parameters drawn from the literature, they find that emissions increase with sprawl and exposure increases with compactness, underscoring the difficulty in assessing compactness net benefits. Finally, Zhang et al. [48] show that there is a significant correlation between urban development patterns and PM

_{2.5}concentrations.

#### 1.5. Summary and Research Goals

- Develop a new SIM for commuting trip distribution, based on Tobit regression estimation [49] and including spatial structure variables measured by competing destinations (CD) [5] and intervening opportunity (IO) [6] factors. It is expected that incorporating these factors will better represent commuting behavior and commuting costs.
- Using the Tobit commuting SIM, develop a new commuting cost minimization model that simultaneously allocates target increments in the population and employment to geographical units across a city or metropolitan area under various scenarios of (a) population and employment densities (land consumption per resident and per employee) and (b) land availability in each geographical unit, as determined by the growth boundaries and environmental constraints. The results of this optimization include a minimum commuting cost surface, which is then to be estimated by polynomial regression, with the densities as independent variables.
- Combining the polynomial commuting cost model with estimated land development cost models and synthetic congestion cost models, develop a total cost minimization model to determine the optimal densities under various growth boundary scenarios and various parametric assumptions for the congestion cost functions.
- Use data on a specific U.S. metropolitan area to test the feasibility of the above-methodological goals. This would be a proof-of-concept goal, but is not intended to provide an actual plan for the local authorities of this metropolitan area.

## 2. Data and Methods

#### 2.1. Overview of the Study Area

#### 2.2. Data Sources

#### 2.2.1. CTPP 2000

#### 2.2.2. Property Data

- Real Estate Taxes|Fredericksburg, VA—Official Website (fredericksburgva.gov);
- Stafford County, VA (staffordcountyva.gov);
- Assessment Office, Spotsylvania County, VA; Real Estate, Caroline County, VA;
- Real Estate, King George County, VA (kinggeorgecountyva.gov).

Residential Property | Workplace Property | |||||||
---|---|---|---|---|---|---|---|---|

Jurisdiction | Average Size (Acre) | Average Property Value (USD) | Average Land Value (USD) | Average Building Value (USD) | Average Size (Acre) | Average Property Value (USD) | Average Land Value (USD) | Average Building Value (USD) |

Caroline | 2.3658 | 220,462 | 47,598 | 162,618 | 5.3395 | 576,642 | 144,972 | 304,321 |

Fredericksburg | 0.3525 | 260,831 | 53,914 | 178,740 | 1.2749 | 945,372 | 388,820 | 506,586 |

King George | 3.2624 | 245,225 | 72,054 | 167,082 | 4.5400 | 517,810 | 196,227 | 271,619 |

Spotsylvania | 1.3127 | 147,640 | 65,580 | 82,059 | 14.0769 | 3,717,248 | 3,330,981 | 386,267 |

Stafford | 1.1375 | 372,153 | 102,621 | 269,531 | 3.3182 | 1,241,485 | 479,788 | 761,687 |

FAMPO | 1.4254 | 247,782 | 77,390 | 167,696 | 8.3923 | 2,309,510 | 1,809,006 | 481,376 |

#### 2.3. Variables

#### 2.3.1. Dependent Variable

#### 2.3.2. Independent Variables

#### Group A

- -
- Percentage of workers driving alone from their residence (P_DA_RES);
- -
- Percentage of workers carpooling from their residence (P_CP_RES);
- -
- Percentage of male workers driving alone from their residence (P_MDA_RES);
- -
- Percentage of male workers carpooling from their residence (P_MCP_RES).

- -
- Percentage of residents in sales or service occupations (P_OCC1_RES);
- -
- Percentage of residents in clerical or administrative support occupations (P_OCC2_RES);
- -
- Percentage of residents in manufacturing, construction, or maintenance occupations (P_OCC3_RES);
- -
- Percentage of residents in professional, managerial, or technical occupations (P_OCC4_RES);
- -
- Percentage of male residents in sales or service occupations (P_MOCC1_RES);
- -
- Percentage of male residents in clerical or administrative support occupations (P_MOCC2_RES);
- -
- Percentage of male residents in manufacturing, construction, or maintenance occupations (P_MOCC3_RES);
- -
- Percentage of male residents in professional, managerial, or technical occupations (P_MOCC4_RES).

- -
- Percentage of Hispanic or Latino residents (P_HIS_RES);
- -
- Percentage of White residents (P_WHT_RES);
- -
- Percentage of Black or African American residents (P_BLK_RES).

- -
- Percentage of resident households with an income of USD 75,000 or more in 1999 (P_HINC_RES);
- -
- Median resident household income (MHI_RES);
- -
- Percentage of resident workers with high earnings (USD 50,000+) in 1999 (P_HERN_RES);
- -
- Percentage of resident workers below the poverty level in 1999 (P_POV_RES);

- -
- Percentage of households with self-owned housing (P_OWNSELF_RES);
- -
- Percentage of households with owned housing with and without a mortgage (P_OWN_RES).

#### Group B

- -
- Percentage of employees below the poverty level (P_BlwPov_EMP);
- -
- Mean travel time (MTT_EMP);
- -
- Percentage of workers with low earnings (P_LERN_EMP);
- -
- Percentage of workers that carpool (P_CarPool_EMP).

- -
- Percentage of workers in manufacturing (P_Mfg_EMP);
- -
- Percentage of workers in wholesale trade (P_WhlTrd_EMP);
- -
- Percentage of workers in retail trade (P_RetTrd_EMP);
- -
- Percentage of workers in service industries (P_serv_EMP);
- -
- Percentage of workers in public administration (P_Pub_EMP).

#### Group C

#### Group D

#### 2.4. Statistical and Optimization Methodology

_{i}at the origin i, the employment E

_{j}at the destination j, several socio-economic variables characterizing either i or j (X…, Y…), the distance d

_{ij}, and competing destinations (CD

_{j}) and intervening opportunity (IO

_{i}) variables that characterize the spatial structure. If F

_{ij}is the commuting flow between i and j, a general SIM can be calculated as follows:

_{i}, b

_{j}, and c

_{ij}are the parameters, X

_{i}and X

_{j}are the variables that characterize the origin i and destination j, Z

_{ij}represents the impedance variables (e.g., distance, time and price) and F

_{ij}is the commuting flow.

_{i}and z

_{j}are the population and employment increments allocated to zones i and j, and

**Z**and

**X**are the corresponding vectors. In addition, ULP is the population density (land area per new resident), ULE is the employment density (land area per new employee), and LAND

_{i}is the land available in zone i for new residents and new employees. The parameters ULP and ULE uniformly apply to all the geographical units. However, the model could be easily modified to test for spatially varying density scenarios. If C

_{ij}is the fixed unit commuting cost between i and j, a general total commuting cost minimization model can be as follows:

_{i}

_{ij}= 0 when the right-hand side of constraint (13) is negative. ULP and ULE are the given parameters in the optimization model, but can be varied in the context of scenario analyses. Constraint (14) simply states that the land to be used for new residents and employees in zone I cannot exceed the land available. In the specific case of the FAMPO region with 188 zones (TAZs), this model has 36,097 variables and 35,911 constraints.

TOTAL COST = commuting cost (TCOM) |

+ land development cost (LDC) |

+ congestion cost (TCON) |

## 3. Results

#### 3.1. Tobit Regression

^{TM}procedure QLIM (qualitative and limited dependent variable models) is the outcome of a multi-step exploratory process. The first estimated model (Model 1) involved only three independent variables that appear in most gravity models, which are as follows: population (P), employment (EMP) and distance (D). All the coefficients turned out to be highly significant (p < 0.0001), with the expected positive sign for P and EMP, and the expected negative sign for D, and with R

^{2}= 0.297 and Pseudo-R

^{2}= 0.042. The Pseudo-R

^{2}is defined as follows:

^{2}for limited dependent variable models.

^{2}. All the five variables of this model (Model 2) turned out to be very significant (p < 0.0001), with a negative sign for IO (as expected) and a positive sign for CD, pointing to agglomeration effects at the destination.

^{2}= 0.355, and Pseudo-R

^{2}= 0.121.

^{2}= 0.466, and Pseudo-R

^{2}= 0.239. Model 4 has stronger performance criteria than Model 3, pointing to the non-linear relationship between commuting flows and the variables P, EMP, D, and CD. The more workers that drive alone to their workplace (P_DA_RES), the higher the flow. The magnitude of this variable coefficient is relatively high (33.11). The share of Black citizens within a population (P_BLK_RES) also has a positive impact on flows. The Black population in the region is a highly educated and affluent middle-class community, hence its mobility and likely positive impact on flows. The occupation and industry variables display the expected signs. The more residents with sales or service (P_OCC1_RES) or clerical or administration (P_OCC2_RES) occupations, the larger the commuting flows. These occupations have stronger impacts than the other two occupations. The percentages of workers in manufacturing, wholesale trade, retail trade, fire, service, and public administration occupations all have positive impacts on commuting flows. Wholesale trade (P_WhlTrd_EMP) and public administration (P_Pub_EMP) have the largest coefficients, 90.08 and 89.72, respectively, followed by manufacturing (55.57), retail trade (45.22), finance industries (36.17), and service occupations (22.13). This result is consistent with the existence of large regional distribution centers, such as CVS and UPS, as well as government and military workers.

#### 3.2. Minimizing Commuting Costs in the Allocation of Population and Employment

#### 3.2.1. Scenarios

- ULP: (0.10–0.50) by 0.05 increments
- ULE: (0.05–0.25) by 0.025 increments

#### 3.2.2. Model Formulation

_{ij}is the distance between TAZs i and j, the objective function to be minimized is as follows:

#### 3.2.3. Optimization Results

#### 3.3. Minimizing All Costs in the Allocation of Population and Employment

#### 3.3.1. Overview of Costs

#### 3.3.2. Estimation of the Commuting Cost Surface

#### 3.3.3. Estimation of Land Development Costs

acreage; employees)

^{2}. However, because the ULP and ULE are the basic variables in the commuting flow function TCOM, the land development cost functions were re-estimated, in log–log form, with ULP and ULE as determinants, together with population P_2006 and employment E_2006. As the densities involve the ratios of acreages to population or employment, the same information is embodied in the new formulations. Furthermore, the exponents of P_2006 and E_2006 must be equal to 1 to avoid scale effects with regard to these variables. This homogeneity allows the estimated functions to be applied to any increment in the population and employment. The new regression results are presented in Table 15.

- IR = average mortgage interest rate over 1997~2006 = 6.71% = 0.0671
- N = number of periods = 30 years (normal mortgage payment period)

#### 3.3.4. Congestion Cost Synthetic Functions

#### 3.3.5. Total Development Cost Minimization

- K1 = 0.1, 0.3, 0.5;
- K2 = 0.1, 0.3, 0.5;
- b = 1.0, 3.0, 5.0;
- d = 1.0, 3.0, 5.0.

## 4. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Jurisdiction | Population | % | Employment | % |
---|---|---|---|---|

Caroline | 22,120 | 9.2% | 1945 | 2.3% |

Fredericksburg | 19,275 | 8.0% | 19,760 | 23.2% |

King George | 16,805 | 7.0% | 9912 | 11.6% |

Spotsylvania | 90,405 | 37.5% | 26,521 | 31.1% |

Stafford | 92,460 | 38.4% | 27,059 | 31.8% |

Total | 241,065 | 100.0% | 85,197 | 100.0% |

Flow | % | |||
---|---|---|---|---|

FAMPO | Internal | 7125 | 10.61 | |

TAZ-to-TAZ | 60,023 | 89.39 | ||

Jurisdiction | Caroline | Internal | 1261 | 1.88 |

Jurisdiction-to-Jurisdiction | 203 | 0.30 | ||

Fredericksburg | Internal | 4056 | 6.04 | |

Jurisdiction-to-Jurisdiction | 12,699 | 18.91 | ||

King George | Internal | 4314 | 6.42 | |

Jurisdiction-to-Jurisdiction | 3173 | 4.73 | ||

Spotsylvania | Internal | 15,863 | 23.62 | |

Jurisdiction-to-Jurisdiction | 6377 | 9.50 | ||

Stafford | Internal | 11,469 | 17.08 | |

Jurisdiction-to-Jurisdiction | 7733 | 11.52 | ||

Total Flow | 67,148 | 100% |

Variable | N | Mean | Standard Deviation | Minimum | Maximum |
---|---|---|---|---|---|

F: flow | 3469 | 19.35 | 34.17 | 4.00 | 990.00 |

P: population | 3469 | 2332.27 | 2659.69 | 15.00 | 15,730.00 |

E: employees | 3469 | 1683.79 | 1687.31 | 4.00 | 6415.00 |

D: distance | 3469 | 10.51 | 6.97 | 0.42 | 41.07 |

Variable | N | Mean | Standard Deviation | Minimum | Maximum |
---|---|---|---|---|---|

F: flow | 31875 | 0 | 0 | 0 | 0 |

P: population | 31875 | 1167.99 | 1765.41 | 0 | 15,730.00 |

E: employees | 31875 | 319.25 | 734.31 | 0 | 6415.00 |

D: distance | 31875 | 18.76 | 9.25 | 0.70 | 50.53 |

Variable | N | Mean | Median | Standard Deviation | Minimum | Maximum |
---|---|---|---|---|---|---|

P | 188 | 1282.26 | 625.00 | 1903.91 | 0.0 | 15,730.0 |

P_DA_RES | 188 | 0.7601 | 0.7782 | 0.1498 | 0.0 | 1.000 |

P_BLK_RES | 188 | 0.1525 | 0.1165 | 0.1421 | 0.0 | 0.674 |

P_OCC1_RES | 188 | 0.1618 | 0.1627 | 0.0844 | 0.0 | 0.600 |

P_OCC2_RES | 188 | 0.1858 | 0.1920 | 0.0914 | 0.0 | 0.455 |

P_OCC3_RES | 188 | 0.2150 | 0.2000 | 0.1140 | 0.0 | 0.580 |

P_OCC4_RES | 188 | 0.1769 | 0.1700 | 0.1002 | 0.0 | 0.495 |

P_M_RES | 188 | 0.4981 | 0.4912 | 0.0918 | 0.0 | 0.984 |

P_UNEMP_RES | 188 | 0.0214 | 0.0165 | 0.0256 | 0.0 | 0.150 |

P_MUNEMP_RES | 188 | 0.0189 | 0.0000 | 0.0305 | 0.0 | 0.200 |

P_CP_RES | 188 | 0.1480 | 0.1334 | 0.1031 | 0.0 | 0.700 |

P_MDA_RES | 188 | 0.7538 | 0.7802 | 0.1866 | 0.0 | 1.000 |

P_MCP_RES | 188 | 0.1517 | 0.1303 | 0.1305 | 0.0 | 1.000 |

P_MOCC1_RES | 188 | 0.1232 | 0.1172 | 0.1031 | 0.0 | 0.667 |

P_MOCC2_RES | 188 | 0.0837 | 0.0817 | 0.0696 | 0.0 | 0.400 |

P_MOCC3_RES | 188 | 0.3279 | 0.3094 | 0.1731 | 0.0 | 1.000 |

P_MOCC4_RES | 188 | 0.2047 | 0.2000 | 0.1375 | 0.0 | 0.695 |

P_HIS_RES | 188 | 0.0185 | 0.0000 | 0.0309 | 0.0 | 0.192 |

P_WHT_RES | 188 | 0.7876 | 0.8220 | 0.1699 | 0.0 | 1.000 |

P_DIS_RES | 188 | 0.1429 | 0.1250 | 0.1028 | 0.0 | 0.600 |

P_HINC_RES | 188 | 0.4013 | 0.4006 | 0.2163 | 0.0 | 1.000 |

MHI_RES | 188 | 54,720.88 | 52,675.00 | 18,921 | 0 | 109,770 |

P_HERN_RES | 188 | 0.2071 | 0.2032 | 0.1140 | 0.0 | 0.5052 |

P_POV_RES | 188 | 0.0218 | 0.0112 | 0.0331 | 0.0 | 0.250 |

P_OWNSELF_RES | 188 | 0.1881 | 0.1667 | 0.1385 | 0.0 | 1.000 |

P_OWN_RES | 188 | 0.7895 | 0.8546 | 0.2137 | 0.0 | 1.125 |

P_3VEH_RES | 188 | 0.3406 | 0.3354 | 0.1614 | 0.0 | 0.769 |

P_OCCU_RES | 188 | 0.9131 | 0.9486 | 0.1428 | 0.0 | 1.000 |

HEDU_RES | 188 | 0.0384 | 0.0341 | 0.0381 | 0.0 | 0.388 |

Variable | N | Mean | Median | Standard Deviation | Minimum | Maximum |
---|---|---|---|---|---|---|

EMP | 188 | 453.1755 | 75.0000 | 964.605 | 0.0000 | 6415.0 |

P_Full_EMP | 188 | 0.3321 | 0.3099 | 0.2435 | 0.0000 | 1.0000 |

P_Veh2Plus_EMP | 188 | 0.7334 | 0.8265 | 0.2895 | 0.0000 | 1.0000 |

P_BlwPov_EMP | 188 | 0.0364 | 0.0108 | 0.0698 | 0.0000 | 0.6667 |

MTT_EMP | 188 | 24.712 | 25.000 | 15.369 | 0.0000 | 102.3 |

P_LERN_EMP | 188 | 0.3681 | 0.3631 | 0.2498 | 0.0000 | 1.0000 |

P_CarPool_EMP | 188 | 0.0883 | 0.0809 | 0.0923 | 0.0000 | 0.4000 |

P_Mfg_EMP | 188 | 0.0379 | 0.0000 | 0.0948 | 0.0000 | 0.7500 |

P_WhlTrd_EMP | 188 | 0.0192 | 0.0000 | 0.0480 | 0.0000 | 0.4000 |

P_RetTrd_EMP | 188 | 0.0864 | 0.0106 | 0.1413 | 0.0000 | 1.0000 |

P_serv_EMP | 188 | 0.3687 | 0.3637 | 0.3014 | 0.0000 | 1.0000 |

P_Pub_EMP | 188 | 0.0327 | 0.0000 | 0.0777 | 0.0000 | 0.4427 |

P_Finan_EMP | 188 | 0.0561 | 0.0000 | 0.1341 | 0.0000 | 1.0000 |

Variable | N | Mean | Median | Standard Deviation | Minimum | Maximum |
---|---|---|---|---|---|---|

D | 35,344 | 17.947 | 17.380 | 9.379 | 0.420 | 50.530 |

Parameter | Estimate | Standard Error | t-Value | Approx Pr > |t| |
---|---|---|---|---|

Intercept | −80.267754 | 4.532873 | −17.71 | <0.0001 |

P | 0.011133 | 0.000426 | 26.13 | <0.0001 |

EMP | 0.027680 | 0.001218 | 22.73 | <0.0001 |

D | −3.966485 | 0.158067 | −25.09 | <0.0001 |

IO_E | −0.000324 | 0.000024329 | −13.33 | <0.0001 |

CD_E | 0.000160 | 0.000035343 | 4.53 | <0.0001 |

P_DA_RES | 28.859800 | 3.679491 | 7.84 | <0.0001 |

P_BLK_RES | 9.602796 | 3.023634 | 3.18 | 0.0015 |

P_OCC1_RES | 23.840650 | 6.515631 | 3.66 | 0.0003 |

P_OCC2_RES | 12.881209 | 5.595655 | 2.30 | 0.0213 |

P_OCC3_RES | 16.725051 | 4.991596 | 3.35 | 0.0008 |

P_OCC4_RES | 17.346662 | 5.671378 | 3.06 | 0.0022 |

P_Mfg_EMP | 36.628397 | 4.083194 | 8.97 | <0.0001 |

P_WhlTrd_EMP | 53.338335 | 7.888539 | 6.76 | <0.0001 |

P_RetTrd_EMP | 22.840645 | 3.075172 | 7.43 | <0.0001 |

P_Pub_EMP | 43.681025 | 5.569463 | 7.84 | <0.0001 |

P_Serv_EMP | 19.602157 | 1.794940 | 10.92 | <0.0001 |

P_Finan_EMP | 21.147896 | 2.754040 | 7.68 | <0.0001 |

P2 | −0.000000662 | 3.1856992 × 10^{−8} | −20.78 | <0.0001 |

E2 | −0.000002661 | 0.000000158 | −16.87 | <0.0001 |

D2 | 0.048005 | 0.004843 | 9.91 | <0.0001 |

POPEMP | 0.000002331 | 0.000000113 | 20.68 | <0.0001 |

EMPCD | −0.000000109 | 1.807805 × 10^{−8} | −6.03 | <0.0001 |

R-square | 0.4657 | |||

Pseudo-R^{2} | 0.2394 | |||

Log-likelihood | −20,466 |

Jurisdiction | 2035 | ||
---|---|---|---|

Employment | Population | ||

Caroline County | 14,216 | 47,007 | |

Fredericksburg | 43,679 | 29,852 | |

King George County | 17,821 | 40,744 | |

Spotsylvania County | 62,551 | 236,885 | |

Stafford County | 69,574 | 238,208 | |

Total GWRC (PD 16) | 207,841 | 592,696 | |

2000 (Existing) | Increments | ||

Employment | Population | ΔE | ΔP |

85,197 | 241,065 | 122,644 | 351,631 |

Scenario L2 | To | |||||
---|---|---|---|---|---|---|

ULP = 0.400 | ULE = 0.200 | CR | FR | KG | SF | SP |

From | CR | 1000 | 3830 | 4849 | 7395 | 4223 |

FR | 0 | 961 | 33 | 528 | 802 | |

KG | 9 | 2205 | 5531 | 2891 | 2369 | |

SF | 0 | 4481 | 1575 | 10,991 | 3796 | |

SP | 40 | 5855 | 1545 | 6215 | 10,695 | |

Scenario L2 | To | |||||

ULP = 0.100 | ULE = 0.050 | CR | FR | KG | SF | SP |

From | CR | 29 | 1582 | 382 | 967 | 1463 |

FR | 0 | 812 | 0 | 341 | 562 | |

KG | 0 | 350 | 92 | 20 | 282 | |

SF | 0 | 2016 | 0 | 3251 | 1322 | |

SP | 0 | 2848 | 0 | 1417 | 3936 | |

Scenario L4 | To | |||||

ULP = 0.400 | ULE = 0.200 | CR | FR | KG | SF | SP |

From | CR | 136 | 288 | 1349 | 2304 | 1415 |

FR | 0 | 780 | 48 | 472 | 592 | |

KG | 0 | 813 | 1057 | 1405 | 919 | |

SF | 0 | 3836 | 1146 | 9045 | 2751 | |

SP | 0 | 4886 | 1544 | 5245 | 8266 | |

Scenario L4 | To | |||||

ULP = 0.100 | ULE = 0.050 | CR | FR | KG | SF | SP |

From | CR | 2 | 7 | 0 | 0 | 23 |

FR | 0 | 763 | 0 | 357 | 549 | |

KG | 0 | 0 | 17 | 0 | 0 | |

SF | 0 | 1787 | 0 | 4030 | 1170 | |

SP | 16 | 4038 | 0 | 2075 | 3837 |

Land Scenario | Density Scenario | ||
---|---|---|---|

ULP = 0.400 | ULP = 0.100 | ||

ULE = 0.200 | ULE = 0.050 | ||

L2 | Objective function | 838,777 | 113,647 |

Total flows | 81,819 | 21,671 | |

Average commuting distance | 10.25 | 5.24 | |

L4 | Objective function | 473,283 | 106,347 |

Total flows | 48,294 | 18,671 | |

Average commuting distance | 9.80 | 5.70 |

Normalized | ||||||||||

Land Scenario L2 | ULE | |||||||||

0.050 | 0.075 | 0.100 | 0.125 | 0.150 | 0.175 | 0.200 | 0.225 | 0.250 | ||

ULP | 0.10 | 12.21 | 15.05 | 17.09 | 17.17 | 17.23 | 19.56 | 20.26 | 21.67 | 22.12 |

0.15 | 15.06 | 16.24 | 17.56 | 18.32 | 20.61 | 22.26 | 22.99 | 23.18 | 23.44 | |

0.20 | 26.31 | 27.15 | 28.03 | 28.16 | 28.42 | 28.92 | 29.06 | 29.15 | 29.55 | |

0.25 | 33.14 | 39.99 | 40.34 | 41.25 | 41.32 | 42.06 | 44.21 | 45.06 | 45.30 | |

0.30 | 40.62 | 46.95 | 53.61 | 57.55 | 59.47 | 61.18 | 62.55 | 64.59 | 66.17 | |

0.35 | 48.17 | 60.51 | 66.65 | 71.39 | 79.09 | 81.80 | 85.00 | 87.99 | 91.40 | |

0.40 | 53.13 | 64.22 | 70.87 | 78.52 | 80.79 | 85.27 | 90.12 | 100.00 | ||

0.45 | 63.43 | 76.50 | 80.40 | |||||||

0.50 | ||||||||||

Normalized | ||||||||||

Land Scenario L4 | ULE | |||||||||

0.050 | 0.075 | 0.100 | 0.125 | 0.150 | 0.175 | 0.200 | 0.225 | 0.250 | ||

ULP | 0.10 | 18.96 | 20.82 | 21.69 | 25.55 | 26.88 | 27.71 | 28.01 | 28.72 | 29.08 |

0.15 | 23.10 | 23.41 | 26.63 | 29.99 | 32.82 | 36.03 | 36.15 | 36.26 | 36.39 | |

0.20 | 31.63 | 35.46 | 38.57 | 39.61 | 39.90 | 40.13 | 40.33 | 40.50 | 40.73 | |

0.25 | 39.40 | 40.17 | 41.09 | 41.72 | 42.26 | 42.71 | 43.14 | 43.56 | 43.97 | |

0.30 | 44.12 | 46.64 | 51.31 | 53.13 | 53.93 | 54.71 | 55.55 | 56.36 | 57.20 | |

0.35 | 48.18 | 51.69 | 56.12 | 65.54 | 69.32 | 71.19 | 71.22 | 72.31 | 73.40 | |

0.40 | 49.13 | 56.64 | 64.89 | 71.43 | 80.24 | 83.10 | 84.40 | 85.64 | 87.05 | |

0.45 | 51.15 | 59.24 | 65.18 | 74.43 | 81.43 | 83.41 | 85.23 | 86.67 | 88.28 | |

0.50 | 55.52 | 63.55 | 72.26 | 80.09 | 88.76 | 92.80 | 95.45 | 97.63 | 100.00 |

Variables | Land Development Strategy | |
---|---|---|

L2 | L4 | |

Intercept | 188,109 (1.71) * | 100,497 (2.16) * |

ULP | −3,410,307 (−3.34) ** | −377,350 (−1.07) |

ULE | 2,816,147 (1.89) * | 117,457 (0.17) |

ULP × ULP | 19,994,468 (5.65) ** | 3,921,962 (3.54) ** |

ULE × ULE | −14,772,237 (−1.68) * | 2,136,213 (0.48) |

ULP × ULE | −4,477,808 (−0.93) | 878,521 (0.52) |

ULP × ULP × ULP | −26,345,984 (−6.61) ** | −5,919,163 (−5.02) ** |

ULE × ULE × ULE | 31,053,141 (1.72) * | −3,265,161 (−0.35) |

ULP × ULP × ULE | 24,799,246 (3.94) ** | 9,630,043 (4.90) ** |

ULP × ULE × ULE | −6,222,188 (−0.62) | −13,204,059 (−3.36) * |

R^{2} | 0.987 | 0.983 |

Land Development Cost (Residential) | Land Development Cost (Employment) | ||||||
---|---|---|---|---|---|---|---|

Intercept | 11.218 | 262.90 (<0.0001) | R^{2} 0.85 | Intercept | 10.810 | 93.39 (<0.0001) | R^{2} 0.78 |

LN(P_2006) | 1.000 | Infty (<0.0001) | LN(E_2006) | 1.000 | Infty (<0.0001) | ||

LN(ULP) | 0.014 | 0.34 (0.7311) | LN(ULE) | 0.502 | 10.40 (<0.0001) | ||

RESTRICT | 16.899 | 2.37 (0.0173) | RESTRICT | 98.442 | 5.01 (<0.0001) |

K1 | ||||||||
---|---|---|---|---|---|---|---|---|

0.1 | 0.3 | 0.5 | ||||||

K2 | b | d | ULP | ULE | ULP | ULE | ULP | ULE |

0.1 | 1.0 | 1.0 | 0.1000 L | 0.0500 L | 0.1000 L | 0.0500 L | 0.1000 L | 0.0500 L |

3.0 | 0.1000 L | 0.0805 | 0.1000 L | 0.0805 | 0.1000 L | 0.0805 | ||

5.0 | 0.1000 L | 0.2211 | 0.1000 L | 0.2211 | 0.1000 L | 0.2211 | ||

3.0 | 1.0 | 0.1447 | 0.0500 L | 0.1930 | 0.0500 L | 0.2206 | 0.0500 L | |

3.0 | 0.1443 | 0.0805 | 0.1917 | 0.0803 | 0.2186 | 0.0801 | ||

5.0 | 0.1435 | 0.2213 | 0.1876 | 0.2212 | 0.2121 | 0.2209 | ||

5.0 | 1.0 | 0.3135 | 0.0500 L | 0.5000 U | 0.0500 L | 0.5000 U | 0.0500 L | |

3.0 | 0.3089 | 0.0793 | 0.5000 U | 0.0763 | 0.5000 U | 0.0763 | ||

5.0 | 0.2952 | 0.2194 | 0.3594 | 0.2174 | 0.3989 | 0.2160 | ||

0.3 | 1.0 | 1.0 | 0.1000 L | 0.0500 L | 0.1000 L | 0.0500 L | 0.1000 L | 0.0500 L |

3.0 | 0.1000 L | 0.1108 | 0.1000 L | 0.1108 | 0.1000 L | 0.1108 | ||

5.0 | 0.1000 L | 0.2500 U | 0.1000 L | 0.2500 U | 0.1000 L | 0.2500 U | ||

3.0 | 1.0 | 0.1447 | 0.0500 L | 0.1930 | 0.0500 L | 0.2206 | 0.0500 L | |

3.0 | 0.1439 | 0.1108 | 0.1905 | 0.1105 | 0.2168 | 0.1103 | ||

5.0 | 0.1435 | 0.2500 U | 0.1872 | 0.2500 U | 0.2113 | 0.2500 U | ||

5.0 | 1.0 | 0.3135 | 0.0500 L | 0.5000 U | 0.0500 L | 0.5000 U | 0.0500 L | |

3.0 | 0.3051 | 0.1090 | 0.5000 U | 0.1042 | 0.5000 U | 0.1042 | ||

5.0 | 0.2932 | 0.2500 U | 0.3551 | 0.2500 U | 0.3917 | 0.2500 U | ||

0.5 | 1.0 | 1.0 | 0.1000 L | 0.0500 L | 0.1000 L | 0.0500 L | 0.1000 L | 0.0500 L |

3.0 | 0.1000 L | 0.1285 | 0.1000 L | 0.1285 | 0.1000 L | 0.1285 | ||

5.0 | 0.1000 L | 0.2500 U | 0.1000 L | 0.2500 U | 0.1000 L | 0.2500 U | ||

3.0 | 1.0 | 0.1447 | 0.0500 L | 0.1930 | 0.0500 L | 0.2206 | 0.0500 L | |

3.0 | 0.1438 | 0.1285 | 0.1899 | 0.1283 | 0.2158 | 0.1280 | ||

5.0 | 0.1435 | 0.2500 U | 0.1872 | 0.2500 U | 0.2113 | 0.2500 U | ||

5.0 | 1.0 | 0.3135 | 0.0500 L | 0.5000 U | 0.0500 L | 0.5000 U | 0.0500 L | |

3.0 | 0.3031 | 0.1265 | 0.5000 U | 0.1205 | 0.5000 U | 0.1205 | ||

5.0 | 0.2932 | 0.2500 U | 0.3551 | 0.2500 U | 0.3917 | 0.2500 U |

K1 | ||||||||
---|---|---|---|---|---|---|---|---|

0.1 | 0.3 | 0.5 | ||||||

K2 | b | d | ULP | ULE | ULP | ULE | ULP | ULE |

0.1 | 1.0 | 1.0 | 0.1000 L | 0.0500 L | 0.1000 L | 0.0500 L | 0.1000 L | 0.0500 L |

3.0 | 0.1000 L | 0.0810 | 0.1000 L | 0.0810 | 0.1000 L | 0.0810 | ||

5.0 | 0.1000 L | 0.2218 | 0.1000 L | 0.2218 | 0.1000 L | 0.2218 | ||

3.0 | 1.0 | 0.1476 | 0.0500 L | 0.2064 | 0.0500 L | 0.2412 | 0.0500 L | |

3.0 | 0.1470 | 0.0809 | 0.2048 | 0.0807 | 0.2387 | 0.0806 | ||

5.0 | 0.1474 | 0.2223 | 0.2030 | 0.2227 | 0.2346 | 0.2228 | ||

5.0 | 1.0 | 0.3399 | 0.0500 L | 0.4456 | 0.0500 L | 0.5000 U | 0.0500 L | |

3.0 | 0.3351 | 0.0801 | 0.4273 | 0.0794 | 0.5000 U | 0.0787 | ||

5.0 | 0.3248 | 0.2226 | 0.3991 | 0.2220 | 0.4423 | 0.2215 | ||

0.3 | 1.0 | 1.0 | 0.1000 L | 0.0500 L | 0.1000 L | 0.0500 L | 0.1000 L | 0.0500 L |

3.0 | 0.1000 L | 0.1108 | 0.1000 L | 0.1108 | 0.1000 L | 0.1108 | ||

5.0 | 0.1000 L | 0.2500 U | 0.1000 L | 0.2500 U | 0.1000 L | 0.2500 U | ||

3.0 | 1.0 | 0.1476 | 0.0500 L | 0.2064 | 0.0500 L | 0.2412 | 0.0500 L | |

3.0 | 0.1466 | 0.1107 | 0.2037 | 0.1106 | 0.2368 | 0.1104 | ||

5.0 | 0.1480 | 0.2500 U | 0.2036 | 0.2500 U | 0.2351 | 0.2500 U | ||

5.0 | 1.0 | 0.3399 | 0.0500 L | 0.4456 | 0.0500 L | 0.5000 U | 0.0500 L | |

3.0 | 0.3316 | 0.1097 | 0.4170 | 0.1089 | 0.4783 | 0.1081 | ||

5.0 | 0.3243 | 0.2500 U | 0.3973 | 0.2500 U | 0.4391 | 0.2500 U | ||

0.5 | 1.0 | 1.0 | 0.1000 L | 0.0500 L | 0.1000 L | 0.0500 L | 0.1000 L | 0.0500 L |

3.0 | 0.1000 L | 0.1281 | 0.1000 L | 0.1281 | 0.1000 L | 0.1281 | ||

5.0 | 0.1000 L | 0.2500 U | 0.1000 L | 0.2500 U | 0.1000 L | 0.2500 U | ||

3.0 | 1.0 | 0.1476 | 0.0500 L | 0.2064 | 0.0500 L | 0.2412 | 0.0500 L | |

3.0 | 0.1465 | 0.1281 | 0.2032 | 0.1280 | 0.2360 | 0.1279 | ||

5.0 | 0.1480 | 0.2500 U | 0.2036 | 0.2500 U | 0.2351 | 0.2500 U | ||

5.0 | 1.0 | 0.3399 | 0.0500 L | 0.4456 | 0.0500 L | 0.5000 U | 0.0500 L | |

3.0 | 0.3299 | 0.1272 | 0.4126 | 0.1262 | 0.4676 | 0.1254 | ||

5.0 | 0.3243 | 0.2500 U | 0.3973 | 0.2500 U | 0.4391 | 0.2500 U |

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## Share and Cite

**MDPI and ACS Style**

Lee, D.J.-H.; Guldmann, J.-M.
Optimal Regional Allocation of Future Population and Employment under Urban Boundary and Density Constraints: A Spatial Interaction Modeling Approach. *Land* **2023**, *12*, 433.
https://doi.org/10.3390/land12020433

**AMA Style**

Lee DJ-H, Guldmann J-M.
Optimal Regional Allocation of Future Population and Employment under Urban Boundary and Density Constraints: A Spatial Interaction Modeling Approach. *Land*. 2023; 12(2):433.
https://doi.org/10.3390/land12020433

**Chicago/Turabian Style**

Lee, David Jung-Hwi, and Jean-Michel Guldmann.
2023. "Optimal Regional Allocation of Future Population and Employment under Urban Boundary and Density Constraints: A Spatial Interaction Modeling Approach" *Land* 12, no. 2: 433.
https://doi.org/10.3390/land12020433