An empirical study of collaborative capacity evaluation and scheduling optimization for a CPD project

In a collaborative product design project, reasonable resource allocation can shorten the development cycle and reduce cost. Team capacity evaluation and a task-team scheduling model are presented. A collaborative team capacity model is constructed, and a 2-tuple linguistic method is used to evaluate the capacity of collaborative teams. Next, the matching degree between design task and collaborative team is defined. A collaborative product design scheduling model considering task-team matching is developed. Combined with the simulated annealing operator, based on the single-coding strategy, self-adaptive multi-point cross and mutation, an improved genetic algorithm is proposed to solve the model. Finally, a case study is presented to validate the method.


Introduction
With the increasing global competition and growing complexity of products, the division of labour is becoming increasingly specialized. As a result, the core firm requires joint development involving customers, suppliers and research institutes to overcome these limitations. Through cross-organizational collaborative product design, it can realize the maximization of resource integration and knowledge sharing as well as the improvement of design efficiency. However, in the process of collaborative product design (CPD), the diversity of design agent and interdependence and mutual restriction between tasks make the collaborative product design process quite complicated. Therefore, design task and resource should be reasonably allocated to shorten the development cycle and reduce cost.
There is a great amount of research work on the task and resource allocation of a collaborative design project. Some of these research studies focused on task identification, task relationship analysis and task scheduling based on Petri Nets [1] and Design Structure Matrix (DSM) [2][3]. Other research studies focused on the establishment of a task and resource dynamic scheduling optimization model and a model solution based on heuristic algorithm [4]  skill level and interests, i.e., each team has its own special abilities and resources. Therefore, collaborative product design requires not only reasonable design task decomposition but also reasonable matching between innovation team and task, such matching has important influence on the efficiency and cost of product design. Capacity reflects the skill or ability sets necessary for the relevant tasks. The capacity model requires a description of the capacity elements for a task. When finding an appropriate team to conduct a design task, team capacity should be considered. For collaborative work, information sharing, goal congruence, decision synchronization, resource sharing, collaborative communication, and joint knowledge creation are significant and interconnecting elements [19][20]. Moreover, they are the prerequisite elements. Thus, the capacity model of a collaborative team is constructed as shown in Fig 1. In the model, the basic resources of a collaborative product design team are information resources, hardware and software resource, brand resource and social net resource. The information resource includes available technical information and industry information. Important customers, government, and partners in other industries constitute the team's social net resource. The comprehensive capacity consists of team learning capability, communication capability, team executive capability, technical capability, service consciousness, and management capability. Learning capability and communication capability are more important than the others at this level. The core capacities are team innovation capability and collaboration capability. Team collaboration requires good communication and executive ability as well as excellent team management. Learning capability and technical capability are important prerequisites and serve as the foundation for innovation. Finally, high efficiency and high quality are the ultimate goals. An empirical study of collaborative capacity evaluation and scheduling optimization for a CPD project

Team capacity evaluation based on the 2-tuple linguistic method
For capacity evaluation, the common methods are based on fuzzy mathematics theory, such as AHP and triangular fuzzy numbers. However, in these methods, fuzzy operation based on the extension principle increases the fuzziness of the results and causes information loss or distortion. In addition, evaluation experts often adopt natural language to express their preference, e.g., they use ''high", ''average" and ''low" to evaluate the team capacity, or they use ''very high", ''high", ''average", ''low" and ''very low" to express their evaluation results. In other words, different experts can express their evaluation information at different levels of granularity. The 2-tuple linguistic method can effectively aggregate natural language evaluation information of different levels of granularity to avoid information loss and make the result more precise [21][22]. Thus, the 2-tuple linguistic method is adopted to evaluate the competencies of the collaborative team.
The specific evaluation steps are as follows: Step 1. The experts with the same granularity are divided into a group. The weight evaluation result of expert k(k = 1, 2,. . .,t) for capacity is denoted as (c y k , b y k ). The evaluation result of team j for task i in capacity given by expert k is denoted as (c y kij , b y kij ). According to Eq (4), the integrated information of group with the same granularity, denoted as ðs y ij ;ã y ij Þ, is obtained.
Step 2. Obtain the weight vector l ¼ ðl 0 1 ; l 0 2 ; . . . ; l 0 u Þ according to the improved EOWA operator, and then, aggregate the integrated information ðs y ij ;ã y ij Þ according to Eq (5) to obtain the comprehensive evaluation information of team j for task i in capacity y, denoted as (s y ij , a y ij ). Next, the weight vector is converted into a crisp value g y ij .

Matching degree between task and team
The matching degree refers to measure of fitness between elements. For example, when matching a project task with the collaborative team, if the matching degree is too low, then the collaborative team's capabilities and resources are not adequate to allow them to complete the task. A higher matching degree ensures that the team can accomplish the tasks high-efficiency and high-quality, but it also means higher cost. To address this trade-off, this paper constructs a task-team matching degree model of collaborative product design project. The task-team matching degree model is constructed in two ways: one is based on the personnel capability matching degree of collaborative team (the comprehensive capacity and core capacities in the capacity model), and the other is based on the available resources matching degree (the basic resources in the capacity model).
The matching degree between task i and team j at the dimension of personnel capabilities, denoted as TC ij , is defined as follows: where p denotes the pth personnel capability, α i p is the weight of the pth personnel capability for task i, g p ij is the evaluation value of the pth personnel capability of team j for task i, and e p i is the required value of the pth personnel capability for task i. In Eq (7), if g p ij >e p i , then take "+"; otherwise, take "-".
Some available resources can be quantified, such as hardware and software. Thus, the matching degree calculation model between project task i and collaborative team j at the dimension of available resource, denoted as TR ij , is defined as follows: where r denotes the rth resource, b r i is the weight of the rth resource for taski, g r ij is the available amount of the rth resource of team j for task i, and e r i is the required amount of the rth resource for task i.
Furthermore, the matching degree (MD ij ) between task i and team j is defined as: where w i1 and w i2 are the weights of the personnel capability and the available resource for task i, respectively.

Scheduling model
In a collaborative innovation project, through rational resource selection and configuration according to the project tasks' requirement, optimal duration and cost are achieved. Parameters: PT: the project duration; For collaborative product design, the shortened duration often leads to increased costs. Chen et al. [25] proposed a linear relationship between the activity time reduction and the cost increases to transfer the time-cost trade-off problem into a linear programming problem. Thus, the optimization objective is as follows: Constraints: In the objective function f(x), a 1 and a 2 are the weights of project duration and cost, respectively. Constraint (12) expresses the resource constraint. Constraint (13) ensures that task i is only performed by one collaborative team. Constraint (14) ensures that one collaborative team can only perform one task at a period, A t denotes the collection of tasks that are conducted at time t. Constraint (15) is the time constraint, and B (q) is the precedence activities setoff task q. Eq (16) is the time taken for collaborative team j to finish task i while considering the matching degree.

The improved GA
The issue proposed in this paper is a combinatorial optimization problem. However, it is different from traditional combinatorial optimization problems because the encoding cannot be repeated. A collaborative team can execute several tasks as long as the tasks do not overlap in one period. To solve the problem, the genetic algorithm is improved, where genetic operators are used to represent the individual of feasible solution in the encoding process. Single-coding in the solution space not only eliminates the decoding process between gene space and solution space but also can enhance the accuracy and reduce the complexity of computation process.
The steps of improved genetic algorithm are as follows: 1. Coding Adopting decimal single coding, each gene locus represents the task code, and the number on the gene locus represents the corresponding matching collaborative team, as shown in Fig 2. 2. Fitness function The fitness function of GA is known as the evaluation function; it is used to determine the quality of individual. In this paper, the objective function is set as fitness function F(x).

Selecting the initial population
Randomly generate a certain number of individuals. Next, remove the repeated individuals and the individuals who do not meet the constraints, choose the best individual into the initial population and select a-1 individual from the remaining individuals randomly. These individuals compose an initial population with number of a. The probability (p i ) that can be selected is set as follows:

Crossover operator
Multi-point crossover is adopted. In the process of evolution, if the current individual fitness is lower than the average fitness, then the individual evolution is invalid. To improve the search speed, it is necessary to improve individual crossover probability. Therefore, the adaptive crossover probability strategy is adopted. The crossover probability(p c ) is defined as where F av and F max are the average fitness and the largest fitness, respectively.

Mutation operator
Execute mutation operation for each individual, the gene changes at a certain probability and varies from 1 to n (n is the total number of collaborative team). In the process of mutation, single point mutation is used the first half of the individual, and multi-point mutation is adopted in the second part.

Selection operator
The previous generation population, the population after crossover and the population after mutation constitute the selection set. Remove the individuals of the population that do not meet the constraints. Next, the best individuals of the preceding generation population, crossover population and mutation population are retained. For the remaining individuals, two individuals are selected randomly, and one of them is chosen using the simulated annealing operator with probability exp (-Δc/θ) to bring into the next generation, and the other is taken back.
Repeat the above procedure until the number of the next generation reaches a, and then go to the next round.

Termination condition, output the optimal
When meet one of the conditions, the iteration is stopped: 1. Fitness of the best individual and the group are no longer rising; 2. The number of iterations reaches the preset number.
The procedure of improved genetic algorithm is shown in Fig 3.

Case study
First, we conducted an experiment on our scheduling optimization algorithm of mobile phone collaborative product design. The relationship of design task is shown in Fig 4. A total of 15 tasks were included in the project, and 20 collaborative teams were available. Standard time and the maximum shorten time of the tasks are shown in Table 1.  The matching degrees between the collaborative teams and the tasks are shown in Tables 2  and 3.
The task costs are listed in Tables 4 and 5.
The parameter configurations of the improved GA were as follows: the initial population size was 20, P c1 was 0.85, P c2 was 0.65, the mutation probability was 0.9, the maximum number of iteration was 800,a 1 was 0.6, and a 2 was 0.4. Based on the data above, the procedures of the improved genetic algorithm were written in Matlab and run on a PC with an Intel Core 2.4 GHz CPU, 4GB RAM, the optimal programme is shown in Table 6.
Under this matching programme, the objective optimal value is 74.10, the duration is 45.7days and the cost is 1,180,000 RMB. The solution obtained by GA is {1 19 4 1 7 8 12 2 19 13 2 1 5 12 2}. The fitness curve of the improved GA and that of the traditional GA are shown Table 2. Matching degree between collaborative teams (G 1 -G 10 ) and tasks (T 1 -T 15  An empirical study of collaborative capacity evaluation and scheduling optimization for a CPD project

Conclusions
In this paper, a competence evaluation and a scheduling model of collaborative product design were studied based on matching degree. In the competence model, the collaborative team capacity is composed of core competency, basic competency and basic resource. Variable competencies or resources have different effects on the matching degree. The 2-tuple linguistic method was used to avoid information loss and make the evaluation result more precise. The scheduling model considering matching degree was established considering matching degree, project duration and cost. In the improved algorithm, single-coding strategy, multi-point mutation and crossover are adopted. Although the case study demonstrated that the proposed approach is a useful tool to obtain the reasonable programme, there are still limitations in the approach, such as the subjectivity of evaluation and the precision of resource quantization. Furthermore, during the process of  collaborative product design, there may be resource conflicts and partner selection conflicts.
In the future, more work on the encouragement and collaboration mechanism for collaborative design should be performed.