## 1 Introduction  
The product service system (PSS) is a design approach that combines products with services, which is widely adopted in the transformation of manufacturing enterprises to enhance product competitiveness [1, 2]. By considering the multidimensional nature of PSS design, the PSS has the potential to create synergies between user needs and corporate profits that cannot be obtained by focusing only on the service performance (e.g., sustainability, serviceability) or more traditional objectives (e.g., time, quality) in product development [3]. Furthermore, PSS provides customers with integrated solutions of products and services, and provides personalized services according to different user needs, so as to improve customer satisfaction [4]. The PSS design cycle comprises a large number of design factors and data [5], which can be used to identify customer requirements and evaluate PSS conceptual solutions. However, excessive focus on customer demands may lead to designs that are too expensive to be implemented as actual industrial products. For example, there are conflicting needs between users' service performance of the product and the product lifecycle cost of the enterprises, which may increase the burdens for enterprises to improve the serviceability of PSS design. For this reason, to satisfy the customer service needs of the product in the PSS design and not exceed the economic capacity of the enterprise, PSS concept trade-off decisions need to be made to ensure the feasibility of downstream activities of PSS design. This research aims to find the optimal trade-off between the economic needs of the company and customer value (CV), so as to solve the optimal ratio between CV and economic objectives of PSS design is still one of the most significant questions needing to be answered. The early-stage evaluation of PSS conceptual solutions involves a conventional multiple-criteria decision analysis problem [6; 7], which includes conventional objectives such as quality, time, cost and performance, and PSS service characteristics. Therefore, the evaluation of PSS designs differs from conventional product evaluations. To select the optimal PSS solution, it is necessary to have accurate evaluation criteria weights and holistically consider the value of each PSS conceptual solution across all criteria. A plethora of weighting models have been used to calculate evaluation criteria weights, such as the analytical network process (ANP) [8], analytic hierarchy process (AHP) [9], and quality function deployment (QFD) [10]. These methods are based on the matrix construction method, and the decision-makers (DMs) can make subjective judgment among the criteria to obtain the subjective weight of these criteria [1]. However, because pairwise comparisons by DMs are necessary to determine the priority of each evaluation criterion, the weightings will contain uncertainties, owing to subjective factors such as the knowledge, skill, and ability of the DMs. Accordingly, attempts have been made to minimize subjectivity, using subjective weighting models, such as the entropy weight method (EWM) [11] and CRITIC method [12]. However, because objective and subjective weighting methods ultimately rely on the evaluation data provided by DMs, their evaluation data inherently contain some ambiguity and subjectivity, which will affect the accuracy of the final decision. Rough sets are often used to capture assessment information from multiple DMs, as they can characterize the uncertainty of the assessment information using interval boundaries, without adopting auxiliary data or membership functions [13; 14]. This helps to ensure the objectivity of the decision-making process. However, aggregating the assessments for the conceptual PSS solution remains a major challenge. During the design of a PSS, each PSS design (solution) is evaluated by numerous stakeholders based on various factors, including functionality, environmental, and social factors.