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2D woven auxetic fabrics (AF) developed by weave design modification provide a continuous, commercial, and low-cost manufacturing process for high strength and stable auxetic structures. However, AFs developed by previous methods have a highly folded surface, which is problematic in printing, laminating, and stitching. For the first time, we studied the auxeticity in a few yarn-based auxetic structures, which provides a uniform surface texture. Reentrant honey-comb geometry was chosen to fabricate auxetic woven fabrics, as such geometry shows the best auxetic nature. Different parameters of auxetic geometry were optimized based on their mechanical response using the statistical tool: gray relational analysis (GRA). The mechanical properties of AFs were compared with conventional (3/1 twill woven) non-auxetic fabric (NAF), having the same specifications. The smaller unit cell (3E) with a reorientation area of 4P showed optimum auxeticity and mechanical properties. The optimized auxetic structure showed superior mechanical properties as compared to conventional fabric. GRA grade of optimized AF was 67.3% higher than the NAF. Most importantly, the tensile strength was 23% higher for optimized AF structure. The bidimensional energy dissipation ability of the AF provides a key advantage in the superior mechanical response. [ABSTRACT FROM AUTHOR]