Complex engineering system optimization usually involves multiple projects or tasks. On the one hand, dependency modeling among projects or tasks highlights structures in systems and their environments which can help to understand the implications of connectivity on different aspects of system performance and also assist in designing, optimizing, and maintaining complex systems. On the other hand, multiple projects or tasks are either happening at the same time or scheduled into a sequence in order to use common resources. In this paper, we propose a dynamic intelligent decision approach to dependency modeling of project tasks in complex engineering system optimization. The approach takes this decision process as a twostage decisionmaking problem. In the first stage, a task clustering approach based on modularization is proposed so as to find out a suitable decomposition scheme for a largescale project. In the second stage, according to the decomposition result, a discrete artificial bee colony (ABC) algorithm inspired by the intelligent foraging behavior of honeybees is developed for the resource constrained multiproject scheduling problem. Finally, a certain case from an engineering design of a chemical processing system is utilized to help to understand the proposed approach.
Nowadays, complex engineering projects or design processes with long development times usually involve multiple disciplines and a great deal of effort. In general, the difficulties in design or development do not only simply arise from engineering complexity but also lie in the organizational sophistication necessary to manage this design or development process. Therefore, it is very important to optimize the design or development process. Usually, a complex system includes a large number of tasks or subprojects, and complex dependencies existing among tasks will cause resource competition and coordination. It means that in order to understand the implications of connectivity on different aspects of system performance, it is necessary to model the task dependencies and then sequence all project tasks to reveal the underlying structure of the design or development process.
In most of the literatures, graphs or directed graphs such as flow charts and signal flow diagrams are used to analyze complex processes. For example, digraphs are easy to assimilate until they become quite large and lose their intuitive [
However, these researches mentioned earlier only considered one aspect of complex engineering system optimization and neglected connections between task dependencies and task sequencing. Due to these reasons, in this work, we propose a dynamic intelligent decision approach to dependency modeling of project tasks in complex engineering system optimization. It takes this decision process as a twostage decisionmaking problem, where in the first stage, large interdependent tasks are decomposed into smaller and manageable task groups by transforming the binary form of task relationships into the quantifiable numerical one. In this stage, three steps are needed. Firstly, DSM is adopted to model dependencies between tasks. And then, a twoway comparison scheme is used to transform the binary DSM into numerical one. Secondly, task clustering based on indices including bid value as well as coordination cost function is realized so as to discompose the largescale project into some subprojects. Finally, a method for solving task clustering is also developed. In the second one, according to the result of task clustering developed in the first stage, the resource constrained multiproject scheduling model is built firstly, subsequently, a discrete artificial bee colony (ABC) algorithm inspired by the intelligent foraging behaving of honeybee is designed for solving this problem. The whole flow chart of this dynamic intelligent decision approach is shown in Figure
The flow chart of the dynamic intelligent decision approach.
This paper is organized as follows. Firstly, the approach to project task clustering in complex engineering system is proposed in Section
Optimization process in a complex system faces the difficulties not only from technology complexity but also from time pressure. Generally, to decompose large interdependent task groups into small ones by modularization is a very efficient approach to realize engineering optimization. This process usually includes three steps, that is, task dependency modeling, task clustering, and problemsolving strategies.
In this paper, the taskbased modeling method is adopted. Here, the design structure matrix representation of the design process is chosen for three reasons. Firstly, it overcomes the size and visual complexity of many graphbased techniques such as PERT and CPM. Secondly, matrices are easy to manipulate and store in computer. Finally, the DSM modeling has been proven by a number of researchers as a useful tool in task scheduling and management [
In general, a simple DSM displays the relationships between components of a system in a compact, visual, and analytical advantageous format. Specifically, in DSM, each row and its corresponding column are identified with the identical labels. Along each row, the marks indicate what other elements the element in that row depends on. A certain column indicates what other elements the element in that row provide to. Diagonal elements do not convey any meaning at this point. Thus, in Figure
Sample of the DSM.
In the engineering system optimization process, the typical clustering process is comprised of two sections: one is to define modules it includes, and the other is to define the relationships among different modules which will realize the whole function of a system together. Consider the implications of system performance as well as the characteristics of the DSM, there are two steps needed to model task dependencies: (1) transforming the binary format of task relationships into the quantifiable numerical ones so as to represent dependency strength among project tasks from viewpoints of structure, function, shape, and so on; (2) building up multidimension DSM model and then normalize these multidimension data.
Dependencies among tasks are determined by their function, structure, shape, and so on. However, the original DSM is populated with “ones” and “zeros” or “
(1) Select a criterion for every pair of tasks compared and compare which one provides more information input to downstream tasks or which one receives more information output from upstream tasks.
(2) Construct a pairwise comparison matrix as follows:
(3) In order to obtain the relative connection measures between the related tasks in rows and in columns, an eigenvector is calculated for each pairwise comparison matrix. This eigenvector denotes the ranking for each comparing task within the comparison matrix.
(4) There are totally
Numerical DSM model obtained from Section
The main purpose of task clustering is to make tasks with close connection in the same block so as to form an independent project blocks, while the ones with loose connection will be in different blocks. In doing so, it is easy to realize optimization in engineering system.
At present, there are many methods used to realize task clustering such as similar coefficient method, ranking method, and path search method. However, it is not satisfactory when all these methods are applied to clustering of DSM, especially that group size is unknown in advance. Due to this reason, other researchers developed more efficient methods, where one of the typical ones is suggested by Thebeau [
Another index presented by Thebeau [
In general, there exist
Initialize the problem, where every task is taken as an independent group.
Randomly select one of tasks
Delete empty groups, subgroups, and same ones.
Choose a new task to repeat the process mentioned above until all tasks has been traversed and the system reaches a stable state.
In the above sections, tasks that have more information relationships will be converged to the same group. Nevertheless, how to arrange these tasks is another important problem to realize engineering system optimization. As a result, a multiproject tasks scheduling problem is proposed in this section. Generally, scheduling process involves allocation of the given resource to projects to determine the start and completion time of the detailed tasks. The allocation of scarce resources then becomes a major objective of the problem, and several compromises have to be made to solve the problem to the desired level of nearoptimality [
The problem consists of the number of projects
Task
There are only renewable resources, and nonrenewable ones are not considered.
Task preemption is not allowable.
In a multiproject environment, the delay of any project will lead to iterations and alterations of related followup work, so we can assume that the objective is to minimize the completion time of all projects but not a certain project.
Based on these assumptions, the problem and the conceptual model will be described as follows. The considered problem consists of
However, during the scheduling of multiprojects, some tasks will not be performed concurrently due to resource constraints and precedence ones which will also increase the difficulty to solve this problem. In this circumstance, precedence constraints among tasks should be satisfied firstly so as to determine an eligible task set. And then, resource conflicts possibly occurred in this eligible set should be identified in order to decide the task priority values, issued from the select priority rule. Therefore, combined DSM, the following will give a simplifying approach of precedence constraints and the task priority values, respectively: (1) set up DSM clustering model of multiproject based on Section
In this section, the basic artificial bee colony algorithm based on the foraging behavior of honeybees is introduced firstly. Subsequently, the discrete artificial bee colony algorithm used for solving the multiproject scheduling problem is proposed.
In the basic ABC algorithm, the colony of artificial bees contains three groups of bees: employed bees, onlookers, and scouts. Employed bees determine a food source within the neighborhood of the food source in their memory and share their information with onlookers within the hive, while onlookers select one of the food sources according to this information. In addition, a bee carrying out random search is called a scout. In ABC algorithm, first half of the colony consists of the employed bees and the rest half includes the onlookers. There is only one employed bee corresponding to one food source. That is to say, the number of employed bees is equal to the number of food sources around the hive. The position of a food source denotes a possible solution of the optimization problem and the nectar amount of a food source corresponds to the quality (fitness) of the associated solution.
The initial population of solutions is filled with
In order to produce a candidate food position from the old one, the ABC used the following equation:
After all employed bees complete their searches, onlookers evaluate the nectar information taken from all employed bees and choose one of food source sites with probabilities related to its nectar amount. In basic ABC, roulette wheel selection scheme in which each slice is proportional in size to the fitness value is employed as follows:
If a position cannot be improved further through a predetermined number of cycles, then that food source is assumed to be abandoned. The scouts can accidentally discover rich, entirely unknown food sources according to (
There are three control parameters used in the basic ABC: the number of the food sources which is equal to the number of employed bees (SN), the value of
The multiproject scheduling problem is a typical NPhard problem and traditional exact algorithms may cause large computation time and is very difficult to find out the optimal solution. With the last decades, various kinds of optimization algorithms based on swarm intelligence have been designed and applied to functionoptimization, taskallocation, and other problems [
(1)
(2)
(3)
performing one SWAP operator to a sequence;
performing two SWAP operators to a sequence;
performing one INSERT operator to a sequence;
performing two INSERT operators to a sequence;
performing the INVERSE operator to a sequence.
(4)
(5)
In this section, a numerical example derived from an engineering design of a chemical processing system is utilized so as to help to understand the proposed dynamic intelligent decision approach firstly. After that, further analysis and discussions about the effect of task clustering analysis on scheduling schemes as well as the performance of ABC algorithm for solving multiproject scheduling problem are also given.
To demonstrate the effectiveness of our proposed approach, we use 20task binary DSM matrix derived from Su et al. [
Task information for an engineering design of a chemical processing system.
Number  Description of tasks  Resource  Duration (day)  Predecessor  




 
1  Operating structure design  3  0  0  0  6  13, 15 
2  Vessel design  5  3  3  2  12  7, 9 
3  Plant layout/general arrangement  4  2  4  2  5  
4  Shipping design  4  2  2  2  8  1, 15 
5  Structure lifting design  5  0  0  0  4  1, 7, 9, 12 
6  Pressure drop analysis  4  2  2  2  4  8, 15, 17 
7  Process engineering  4  3  3  2  3  
8  Structural documentation  0  4  2  0  3  3, 4, 5 
9  Size valves  2  1  5  3  4  3, 12 
10  Wind load design  4  0  0  0  5  1, 2, 8 
11  Seismic design  5  0  0  0  9  3, 5, 7, 12 
12  Piping design  3  0  0  1  4  
13  Process and instrumentation diagram  3  2  0  1  2  2, 7, 12 
14  Equipment support  3  0  3  2  2  1, 2, 6, 7, 15 
15  Pipe flexibility analysis  4  2  2  2  2  2, 3, 12 
16  Design documentation  5  2  2  0  4  1, 2, 3, 5, 10, 18 
17  Foundation load design  5  2  4  2  5  4, 9, 13 
18  Insulation structural design  2  2  3  3  5  1, 2, 7, 8, 11 
19  Structural bill of material (BOM)  3  2  3  4  6  5, 10, 18 
20  Assembly design  4  0  0  0  7  4, 9, 12, 14 
In the first stage, according to dependency modelling technology mentioned in Section
DSM model of design process.
01DSM
Numerical DSM
Subsequently, using a twoway comparison scheme, we can transform the binary DSM into the numerical one. Here, two criterions named task evolution (represented by EC) and task sensitivity (represented by SC) are adopted to perform pairwise comparisons, where the former means the information transfer rate to the element
The detailed computation process including every pair of tasks compared in DSM are not given due to the length limitation of this paper, and the final computation result is shown in Figure
After dependency modeling based on DSM, a clustering algorithm based on coordination cost mentioned in Section
A flow chart of task clustering algorithm.
The final task clustering result is displayed in Figure
Task clustering result of design process.
The change curve of coordination cost function.
In the second stage, according to the clustering result, the task planning problem can be transformed into multiproject scheduling one. Considering the objective function of the problem is to minimize the delay time of all projects, we must define the shortest makespans of all projects in advance, and critical path method (CPM) is used to obtain their values of the shortest makespans (i.e., 5, 18, 28, and 13). Moreover, so as to simplify the mathematic model, we set all projects that have the same weight coefficients. That is to say
Gantt chart of task planning.
We can see from Figure
In this section, the effect of task clustering analysis on scheduling schemes as well as the performance of ABC algorithm for solving multiproject scheduling problem is discussed, and extensive experiments to analyze theses have been illustrated. The project test problems are generated by the project generator ProGen developed by Kolisch et al. [
Problem instances generated by ProGen for the testing problem subset.
Problem subset  NOI 



MP30  20  30  4 
MP60  20  60  4 
MP90  20  90  4 
MP120  20  120  4 
MP150  20  150  4 
MP180  20  180  4 
MP210  20  210  4 
NOI: no. of instances;
In order to analyze the effect of task clustering analysis on scheduling schemes, two indexes are introduced: one is the average deviation of project delay time, and the other is the computation time, where the former is used to measure robustness of search algorithms and the latter to compare the complexity of computation. Table
The effect of task clustering analysis on scheduling schemes.
Problem subset  NOI  Average deviation of project delay time  Computation time (s)  




 
MP30  20  14.3  14.5  0.10  0.22 
MP60  20  21.3  23.4  0.53  0.88 
MP90  20  29.7  36.9  2.14  3.55 
MP120  20  40.2  53.4  3.21  5.71 
MP150  20  58.5  81.3  4.35  8.77 
MP180  20  82.6  114.8  5.94  12.45 
MP210  20  108.7  152.6  7.12  19.88 
According to task clustering result obtained in the first decision stage, the ABC algorithm is used to solve multiproject scheduling problem in the second one. In order to get the most out of the ABC algorithm, parameter setting mentioned in Section
Comparison of performance obtained by different algorithms.
Problem subset  Schedules  ABC  SA  ACO  AIS  

APD  LTM 

APD  LTM 

APD  LTM 

APD  LTM 


1000  21.1  98.6  0.09 



22.4  93.5  0.07  25.6  101.4  0.07  
MP30  5000  19.3  95.3  0.23  19.3  91.3  0.19 



24.3  97.5  0.21 
10000  19.1  89.9  0.67  18.6  91.3 



0.61  22.1  95.1  0.61  
 
1000  42.1  130.7  0.61 

147.4 

42.1  130.4  0.98 


0.62  
MP60  5000 

125.1  3.9  41.5  140.3 

42.0 

3.7  41.9  127.6  3.9 
10000 


7.9  41.5  138.6 

41.8  124.5  8.8  41.8  127.5  9.5  
 
1000  68.5  254.7  2.3  87.4 

0.9 

287.9  1.2  67  296 


MP90  5000 


9.5  81.3  252.4 

65.1  277.6  5.1  66.9  275  4.8 
10000 


23.1  75.6  252 

64.8  274.1  10.3  66.9  251.4  11.3  
 
1000  87.1  345.1  3.1  121.6  345 

150.4  387.9  1.5 


1.34  
MP120  5000 


11.6  119.3  340.1 

144.3  366.4  6.1  86.5  338.1  5.77 
10000 


32.2  115.6  335.7 

140.2  354.2  12.9  86  335.2  14.8  
 
1000  116.9 

4.3  125.4  419.9 

130.7  443.5 


415.3  2.8  
MP150  5000  110.4 

12.9  124.1  415.3  6.9  127.6  430.1 


405.5  8.3 
10000 


40.5  122.3  413.5 

125.5  427.7  14.9  107.4  401.1  19.3  
 
1000  147.1 

5.9  154.7  520.1  3.2  150.9  519.9 


519.4  3.6  
MP180  5000 


15.9  149.1  515.4 

144.3  514.3  8.2  138.4  512.2  10.7 
10000 


51.1  147.1  512.5 

141.8  510.5  21.7  134.9  508.5  29.1  
 
1000 


7.1  191.4  653.1 

199.3  647.9  4.7  189.7  625.1  6.1  
MP210  5000 


19.3  186.6  651.4  11.2  195.1  641.6 

186.9  624.3  15.7 
10000 


67.8  185.1  650.1 

189.8  639.2  24.1  185.1  621.4  58.1 
In this study, we have presented a twostage intelligent decision approach to dependency modelling of project tasks in complex engineering system optimization. Both the dependencies among tasks and sequence of project tasks are clearly identified. In the first stage, in order to optimize the complex engineering system, we further investigate task evolution degree and sensitivity degree based on a twoway comparison scheme. In addition, we have also introduced DSM model that systematically quantifies the strength between related tasks and decomposes large interdependent task groups into smaller and manageable subprojects. Subsequently, according to decomposition results obtained from the first stage, the discrete artificial bee colony (ABC) algorithm is developed for project task planning in the second stage, where DSM has also adopted to simplify the mathematic model of the task planning. In doing so, a better sequence of project tasks is expected because of less communication links and simpler information flows among tasks in different subprojects. The major contributions of the research are as follows: (a) the integrated model of complex engineering system optimization is developed. This has led to a visible dependency structure and a simpler task sequence through systematic procedures; (b) the task dependencies in the complex system are clearly identified by quantitative measures. It is very important to offer a great research potential in understanding and analyzing the implications of connectivity on different aspects of complex system performance; (c) the decomposition of a large number of task groups and the planning of projects tasks lay a sound foundation for engineering optimization. In addition, each subproject obtained from decomposition has been limited to a manageable size so that tasks in the same subproject have more communication links. The results will serve as the task requirements for efficiently scheduling project tasks.
In this work, we focus on dependency modeling of project tasks in complex engineering system optimization. For sound project task management, it is necessary to consider influences from external factors. Due to this reason, our future research extension will focus on analysing the effects from customers’ requirements on engineering optimization process. After that, how to apply this twostage intelligent decision approach to other optimization fields also needs further study.
This research is supported by National Natural Science Foundation of China (Grant nos. 71071141, 71001088, and 71171089), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant nos. 200804870070 and 20103326120001), and Humanity and Sociology Foundation of Ministry of Education of China (Grant no. 11YJC630019).