Summary: While the Taguchi capability index developed by Chan (J Qual Technol 20(3):162-175, 1988) takes the process targeting issue into consideration, it fails to account for processes with asymmetric tolerances, which are common in practice. Thus, Chen (Int J Reliab Qual Saf Eng 6(4):383-398) modified this index to include processes with asymmetric tolerances. This index is an important tool for the assessment of quality characteristics with asymmetric tolerances, which are common in practice. As the probability density function of the index is complex, statistical inference can be fairly difficult for quality or process engineers. Furthermore, sample sizes are often small in practice to increase decision-making efficiency, but this can decrease assessment accuracy. To address this issue, we employed a mathematical programming approach to make it more convenient for quality or process engineers to derive the upper confidence limit of the index. We also adopted the suggestion put forward by previous studies to incorporate historical data or expert experience in confidence-interval-based fuzzy testing. The proposed approach therefore has increased assessment accuracy, is convenient to apply in practice, and meets the need for swift responses.