Renewal reward process for $$T$$T-related fuzzy random variables on $$(\mathbb {R}^{p}, \mathbb {R}^{q})$$(Rp,Rq)

In this paper, following our previous studies, we investigate the renewal rewards process with respect to the necessity, credibility, chance measure and the expected value in which the random inter-arrival times and random rewards are characterized as weighted fuzzy numbers under $$t$$t-norm-based fuzzy operations on $$\mathbb {R}^{p}$$Rp and $$\mathbb {R}^{q}\,\,p,\,q \ge 1,$$Rqp,q≥1, respectively. Many versions of $$T$$T-related fuzzy renewal rewards theorems are proved by using the law of large numbers for weighted fuzzy variables on $$\mathbb {R}^{p}$$Rp. An application example is provided to illustrate the utility of the results.