This work proposes a joint power control and access points (APs) scheduling algorithm for uplink cell-free massive multiple-input multiple-output (CF-mMIMO) networks without channel hardening assumption. Extensive studies have done on the joint optimization problem assuming the channel hardening. However, it has been reported that the channel hardening may not be validated in some CF-mMIMO environments. In particular, the existing Use-and-then-Forget (UatF) bound based on the channel hardening often seriously underestimates user rates in CF-mMIMO. Therefore, a new performance evaluation technique without resorting to the channel hardening is indispensable for accurate performance estimations. Motivated by this, we propose a new bound on the achievable rate of uplink CF-mMIMO. It is demonstrated that the proposed bound provides a more accurate performance estimate of CF-mMIMO than that of the existing UatF bound. The proposed bound also enables us to develop a joint power control and APs scheduling algorithm targeting at both improving fairness and reducing the resource between APs and a central processing unit (CPU). We conduct extensive performance evaluations and comparisons for systems designed with the proposed and existing algorithms. The comparisons show that a considerable performance improvement is achievable with the proposed algorithm even at reduced resource between APs and CPU.