## 1 Introduction  
As the most common production scheduling problem, the resource-constrained project scheduling problem (RCPSP) is widespread in various fields [1, 2, 3]. The RCPSP is meant to schedule activities without preemption under precedence constraints and limited resource availability to optimize the given objectives, such as the make-span and total cost [4, 5, 6, 7, 8]. In realistic situations, both the manpower and multipurpose machine master several skills, and the level of each skill is differentiated [9]. The execution of each activity requires a resource that masters a certain skill at a standard level [10]. Thus, the multi-skill resource-constrained project scheduling problem (MSRCPSP) is proposed to deal with these situations. As an extension of the RCPSP, the MSRCPSP has attracted more and more attention of researchers recently [11]. [PERSON] et al. [6] developed a MSRCPSP with a new heuristic framework to cooperate with resource weight and activity grouping. [PERSON] et al. [12] proposed a genetic programming hyper-heuristic approach to handle the MSRCPSP, where genetic programming as a high-level strategy managed the low-level heuristics. [PERSON] et al. [13] used a function of the starting time and a specific deteriorating date to express the duration of each activity for the MSRCPSP. [PERSON] et al. [14] studied a MSRCPSP with skill switches, where a new encoding scheme was used to reduce search space and a greedy-like decoding scheme was used to reorder activities to reduce skill switches. [PERSON] and [PERSON] [15] proposed a MSRCPSP considering transfer times. [PERSON] et al. [16] addressed a MSRCPSP considering manpower under shift differential payments and equipment with time-of-useelectricity tariffs. [PERSON] and [PERSON] [17] studied a MSRCPSP considering project members' skill factor. In the above studies on MSRCPSPs, most of them assumed that the resource allocation of the activity is normally given in advance and constant over its duration. However, it is often the case that in real projects, the resource allocation for the activity is unknown beforehand; it is generated in real-time according to the process of the scheduling plan. Relevant literature about RCPSPs considering flexible resource allocation is rather scarce. [PERSON] and [PERSON] [18] introduced a concept of the activity's \"resource profile\", in which resource quantities allocated to activities may vary for the entire duration, and the duration of each activity indeed resulted from the resource quantities at each time period during its processing. Based on this, [PERSON] [19] studied a RCPSP with a flexible resource profile in continuous periods. [PERSON] et al. [20] proposed a RCPSP considering a flexible resource profile in discrete time periods. However, there is one thing to point out: the above literature on flexible resource profiles is entirely limited to a single resource, without considering multi-skill resources. Considering the flexible resource profiles in MSRCPSP, it is not only more in line with a real project but also has greater flexibility in generating a scheduling plan, so as to obtain a more perfect scheduling plan. To the best of our knowledge, this paper is the first attempt to present the multi-skill resource-constrained project scheduling problem with flexible resource profiles (F-MSRCPSP). Given the MSRCPSP is an NP-hard problem [21], as an extension of the MSRCPSP, the F-MSRCPSP is more complicated. It not only needs to consider the selection of multi-skill resources, but it must also adjust the resource allocation of each time period for each activity on the basis of the MSRCPSP. In recent years, many meta-heuristics algorithms have been applied to solve RCPSP-like optimization problems [22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. Among them, the multi-objective fruit fly optimization algorithm (MOFOA), as an effective optimization algorithm, has shown promising potential for solving complex engineering optimization problems.