CR+DES: Hybrid Multi-Objective Optimization

• CR+DES algorithm is a hybrid multi-objective optimization method that combines differential evolution’s scaled difference vectors with NSGA-II’s selection mechanisms for urban land-use allocation.
• It balances exploration and exploitation through adjustable scaling factors and systematic constraint relaxation, yielding enhanced performance in complex, nonlinear planning problems.
• Empirical studies reveal that CR+DES achieves significant improvements in land-use compatibility and economic metrics, validated by robust statistical tests in real-world urban scenarios.

The CR+DES algorithm is a customized multi-objective optimization approach that combines scaled difference vector operators from differential evolution with the population management and selection mechanisms of multi-objective genetic algorithms, particularly NSGA-II. Developed within the context of urban land-use allocation, CR+DES targets complex mixed-use planning problems characterized by nonlinear objectives and stringent feasibility constraints, aiming for optimal trade-offs between land-use compatibility and economic value.

1. Algorithmic Integration: Differential Evolution and Multi-Objective Genetic Framework

CR+DES systematically hybridizes the exploratory mutation principles of differential evolution (DE) with the Pareto-based ranking and crowding distance of multi-objective genetic algorithms. Traditional DE performs mutation through the generation of offspring via scaled difference vectors:

$v = x_{r_1} + F \cdot (x_{r_2} - x_{r_3})$

where $x_{r_1}$, $x_{r_2}$, and $x_{r_3}$ are distinct vectors from the current population and $F$ is a scaling factor.

In CR+DES, with a specified probability for each reproduction event, candidate solutions are generated as:

$v = x_i + F \cdot (x_j - x_k)$

where $x_i$ is a current candidate solution and $x_j$, $x_k$ are other randomly selected solutions. This direct injection of DE-style difference vectors within the multi-objective optimization framework enables efficient exploration of a rugged search space, complementing the crossover and mutation mechanics typical of genetic algorithms. Pareto ranking and crowding distance metrics are then applied for population sorting and diversity maintenance.

2. Scaled Difference Vectors and Their Role in Solution Exploration

By leveraging scaled difference vectors, CR+DES achieves enhanced exploration of the feasible region, especially relevant for urban allocation problems where compatibility metrics and economic objectives generate complex landscapes with multiple local optima. Step sizes, regulated via the scaling factor $F$, determine the algorithm’s ability to traverse promising regions:

• Small $F$: promotes local exploitation
• Large $F$: increases diversity, risks random search

The adaptability of $F$ allows CR+DES to balance between convergence towards high-performing areas and maintaining sufficient solution diversity to escape suboptimal local basins.

3. Constraint Relaxation Strategy During Optimization

A distinguishing feature of CR+DES is its systematic constraint relaxation methodology during intermediate search. Constraints, such as proportional area limits and plot change percentages, are explicitly relaxed:

• The allowable area change $\gamma$ may be increased (e.g., from 30% up to 40–100%)
• The percentage $\mu$ of plots allowed to change likewise may be temporarily increased

This expansion of the feasible region enables broader exploration, avoiding premature convergence. Mathematically, area constraint relaxation is represented as:

$$(1 - \gamma) \cdot \sum_{i=1}^N A_{i,m}^{(actual)} \leq \sum_{i=1}^N x_{i,m} F_i \leq (1 + \gamma) \cdot \sum_{i=1}^N A_{i,m}^{(actual)}$$

At final stages, original strict constraints are reimposed, ensuring output solutions are practical for real-world deployment.

4. Mathematical Formulation and Objective Functions

CR+DES operates over multi-objective functions and nonlinear constraints. Core formulations include:

• Compatibility Objective:

$$\text{Compatibility} = \sum_{i \in I} \sum_{j \in J(i)} \sum_{l=1}^K \sum_{m=1}^K C_{l,m} x_{i,l} x_{j,m} F_i F_j$$

• Price Objective:

$$\text{Price} = \sum_{i=1}^N \sum_{m=1}^K P_{i,m} x_{i,m}$$

• Unity Summation Constraint:

$\sum_{m=1}^K x_{i,m} = 1, \quad \forall i \in I$

These problem formulations reflect the nonconvex and combinatorial nature of mixed-use land allocation, requiring multifaceted search and selection strategies provided by CR+DES.

5. Statistical Validation and Empirical Performance

Algorithmic outcomes are subject to robust non-parametric statistical validation. The Kruskal–Wallis test is employed to compare performance across multiple runs and algorithms, followed by post-hoc Bonferroni–Dunn pairwise testing. Results are presented via Compact Letter Display (CLD), where distinct letters denote statistically significant differences.

The CR+DES algorithm consistently achieves significant improvements over state-of-the-art methods:

• Land-use compatibility: 3.16% improvement compared to leading approaches
• Economic objectives: Competitive performance; MSBX+MO (another algorithm) achieves highest price optimization (3.3% improvement), but CR+DES maintains strong Pareto diversity

The CR+DES method reliably produces a broader Pareto front for land-use compatibility and economic value, indicating enhanced solution diversity and dominance.

6. Application to Real-World Urban Planning Problems

In a case study involving 1,290 plots in Dhaka, Bangladesh, CR+DES demonstrates practical applicability, balancing governmental policy requirements against economic and spatial constraints. By integrating scaled difference vector operators and probabilistic constraint relaxation, it provides urban planners with solutions that are both feasible and superior in compatibility, substantiated by rigorous statistical validation.

7. Comparative Context and Implications

While CR+DES finds primary application in urban land-use optimization, its underlying methodological principles connect with broader combinatorial optimization and metaheuristic research. The adoption of differential evolution operators within multi-objective genetic search is supported by statistical evidence as enhancing algorithmic performance, and the relaxation/reimposition paradigm for constraints represents a robust strategy for real-world feasibility alignment. These findings suggest a generalizable approach for other multi-objective, constraint-dominated problems where conventional GAs or DE alone may struggle.

In summary, CR+DES distinctively merges differential evolution’s scaled difference mechanism with NSGA-II’s elitist population sorting, leveraging systematic constraint relaxation to facilitate broad search within nonlinear, complex solution spaces. Empirical results demonstrate its superiority in balancing mixed-objective criteria, establishing its value for optimization scenarios in urban analytics and comparable domains (Aosaf et al., 21 Aug 2025).