A Hybrid Grey Wolf and Dipper-Throated Optimizer for

Engineering Design Optimization

Mohamed Saber 1,* Faustino D. Reyes2

1 Electronics and Communications Engineering Department, Faculty of Engineering, Delta University for Science and Technology,

Gamasa City 11152, Egypt

2 ICT Bahrain Polytechnic, PO Box 33349, Isa Town, Bahrain

Received: May 27, 2026 Revised: July 11, 2026 Accepted: September 02, 2026 ⋆ Corresponding author

ABSTRACT

Optimization plays a fundamental role in engineering design, enabling cost reduction, performance enhancement,

and constraint satisfaction. Metaheuristic algorithms such as the Grey Wolf Optimizer (GWO) and Dipper-Throated

Optimizer (DTO) have been widely used for solving complex optimization problems. However, standalone algorithms

often suffer from premature convergence and limited exploration capabilities, necessitating the development of

hybrid approaches. This chapter introduces a novel hybrid algorithm, GWO+DTO, which combines the exploratory

strength of GWO with the exploitative efficiency of DTO to improve optimization performance. The effectiveness of

GWO+DTO is evaluated on two benchmark engineering problems: the Pressure Vessel Design Problem and the

Tension/Compression Spring Design Problem, comparing its results with standalone GWO and DTO. Experimental

findings demonstrate that the hybrid approach achieves superior performance, obtaining the best cost of 5950.28 in

the pressure vessel problem and 0.01266 in the spring design problem, outperforming the individual algorithms in

accuracy and efficiency. Additionally, GWO+DTO requires fewer function evaluations, highlighting its computational

efficiency. The proposed hybrid method presents a promising alternative for tackling real-world engineering

optimization challenges, with potential applications in multi-objective and large-scale optimization problems.

Keywords: Hybrid Metaheuristic Grey Wolf Optimizer (GWO) Dipper-Throated Optimizer (DTO) Engineering

Optimization Constrained Optimization

1. INTRODUCTION

Optimization is a fundamental aspect of engineering, science,

and technology, playing a crucial role in designing efficient

and cost-effective solutions across diverse fields. Many realworld

problems involve highly complex, nonlinear, multimodal,

and constrained search spaces, making it challenging

to find optimal solutions using traditional mathematical programming

techniques [1, 2]. These classical methods, such

as gradient-based and linear programming approaches, often

require well-defined objective functions, differentiability

conditions, and convexity assumptions, which limit their applicability

in solving practical engineering problems. As a

result, researchers have increasingly turned to heuristic and

metaheuristic optimization techniques, which provide robust

and flexible alternatives for handling complex and large-scale

optimization problems [3, 4].

Metaheuristic algorithms are computational methods that

imitate natural processes to iteratively refine candidate solutions

until an optimal or near-optimal solution is found.

These algorithms are designed to balance two essential aspects

of optimization: exploration (searching across diverse

regions of the solution space) and exploitation (refining

promising solutions in local neighborhoods) [5, 6]. Over