Optimal Testing Effort Allocation for a Software Reliability Growth Model in Fuzzy Environment via Optimal Control Theoretic Approach

Authors

  • Deepika Gaur Department of Mathematics, Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India
  • Pradeep Kumar Department of Mathematics, Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India
  • Kuldeep Chaudhary Department of Mathematics, Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India
  • Shivani Bali Jaipuria Institute of Management, Noida, Uttar Pradesh, India
  • Vijay Kumar Department of Mathematics, Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India

DOI:

https://doi.org/10.13052/jrss0974-8024.1823

Keywords:

Fuzzy constraint, maximum principle, optimal control theory, testing effort allocation, necessity/possibility

Abstract

The available testing resources are usually restricted during the software testing process. It’s usual to presume that these limitations are deterministic when discussing optimal control models. This isn’t true in practice. For instance, a budget for the use of testing resources could be the first step in the testing process. However, it is possible that fault detection is accelerated during the fault testing process to meet unexpectedly high fault counts, which calls for additional funding for testing resources. These enhanced numbers are obviously unknown in nature. In this case, fuzzy set theory is a plausible model in sense of degree of uncertainty and hence the testing resource expenditure constraints i.e. total budget becomes imprecise in nature. By integrating fuzzy logic into the budgetary framework, this approach offers flexibility in decision-making, enabling more realistic and adaptable strategies. The aim of this research is to look into an optimal way to allocate testing resources in order to minimize software costs during the testing and operation phases while taking independent and dependent faults into account. The model is formulated as an optimal control problem for where the budget constraint is expressed as necessity and /or possibility type. The proposed problem is solved for an optimal testing effort policy employing Pontryagin’s Maximum Principle. A numerical example is presented to support the theoretical optimal control model. The values of the optimal testing effort expenditure function are displayed in both tabular and graphic forms.

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Author Biographies

Deepika Gaur, Department of Mathematics, Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India

Deepika Gaur is a promising young researcher serving as a Research Scholar at the Department of Mathematics within the Amity Institute of Applied Sciences at Amity University Uttar Pradesh, Noida, India. She received her MSc degree in Mathematics from Maharishi Dayanand University, Rohtak, India. Her research interest includes software reliability and mathematical modelling.

Pradeep Kumar, Department of Mathematics, Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India

Pradeep Kumar received his MSc in Mathematics from CSJMU Kanpur India in 2016. He has completed his PhD from the Department of Mathematics, Amity University Noida, Uttar Pradesh, India. He has published more than 7 research papers in the areas of mathematical modelling and optimisation, marketing and software reliability in international journals and conferences.

Kuldeep Chaudhary, Department of Mathematics, Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India

Kuldeep Chaudhary is an accomplished academic professional serving as an Associate Professor at the Department of Mathematics, Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India. He has published more than 30 research articles in the International Journals/book chapters/conferences. He is an editorial board member of IJSA, Springer. He is a life member of Society for Reliability Engineering, Quality and Operations Management (SREQOM).

Shivani Bali, Jaipuria Institute of Management, Noida, Uttar Pradesh, India

Shivani Bali is a Professor and Area Chair (Business Analytics) at Jaipuria Institute of Management, Noida. She holds a Master’s and Ph.D. from the Department of Operational Research, University of Delhi, India, and brings over two decades of expertise in teaching, research, corporate training, and consultancy in Artificial Intelligence (AI), Machine Learning (ML), Data Science (DS), and Business Analytics. She has published research articles in many reputed journals indexed in SCI/ WoS/Scopus and has authored two academic and three edited books. She also has three Patents granted in her name by the Government of India.

Vijay Kumar, Department of Mathematics, Amity Institute of Applied Sciences, Amity University Uttar Pradesh, Noida, India

Vijay Kumar is a Professor at Department of Applied Mathematics, Amity University, Noida. He received his MSc in Applied Mathematics and MPhil in Mathematics from Indian Institute of Technology (IIT), Roorkee, India in 1998 and 2000, respectively. He has completed his PhD from the Department of Operational Research, University of Delhi. Currently, he is a Professor in the Department of Mathematics, Amity Institute of Applied Sciences, Amity University, Noida, India. He is co-editor of two book and has published more than 70 research papers in the areas of software reliability, mathematical modelling and optimisation in international journals and conferences of high repute. His current research interests include software reliability growth modelling, optimal control theory and marketing models in the context of innovation diffusion theory. He has edited special issues of IJAMS and RIO journal. He is an editorial board member of IJSA, Springer. He is a life member of Society for Reliability Engineering, Quality and Operations Management (SREQOM).

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Published

2025-08-09

How to Cite

Gaur, D. ., Kumar, P. ., Chaudhary, K. ., Bali, S. ., & Kumar, V. . (2025). Optimal Testing Effort Allocation for a Software Reliability Growth Model in Fuzzy Environment via Optimal Control Theoretic Approach. Journal of Reliability and Statistical Studies, 18(02), 313–342. https://doi.org/10.13052/jrss0974-8024.1823

Issue

2025: Vol 18 Iss 2

Section

Articles