Abstract
In the real-world complex product design (CPD) process, task allocating is an ongoing reactive process where the presence of unexpected design change is usually inevitable. Therefore, reallocating is necessary to respond to design change positively as a procedure to repair the affected task plan. General reallocating literature addressed the reallocating versions with fixed executing time. In this paper, a multi-objective reallocation model is developed with a feasible assumption that the task executing time is controllable. To illustrate this idea, a compressing executing time strategy (CETS) is proposed in CPD process, where the executing time can be controlled with a non-linear compression cost. When design change occurs during the executing, task-resource reallocating is required to absorb the interference effects. Reallocating implies an increase in design cost and system instability; the proposed method CETS can address this issue effectively. CETS considers three objectives: completing time, stability, and change-adaptation cost. An adaptive multi-objective hybrid genetic algorithm and tabu search (AMOGATS) is developed to solve this mathematical method. The computational results of specific simulation examples verify the superiority. It shows that CETS is sensitive to design change, and the proposed algorithm AMOGATS can be effective to achieve the allocating by coordinating the objective consistency.
1 Introduction
Complex product is identified as the core manufacturing tool and the important output product in equipment manufacturing. It has already occupied a dominant position of advanced manufacturing strategy in the world.
However, the quality and social/economic value of complex product is basically determined in the design process. Microscopically, complex product design (CPD) process can be divided into a series of design phases with constrained relationship; each phase includes multiple sub-processes. The guarantee that all design phases and sub-processes are under supervision is a key sticking point for high quality, which is regarded as an important issue in equipment manufacturing industry. Due to the cross relationship in CPD, design tasks have to be performed by designers from different domains with multiple resources. CPD can be deemed a resource-complicated system or a process-complicated system with related elements. The issue around how to allocate resources to tasks properly and effectively in response to design change is an essential activity in CPD process.
Unexpected design change such as resource unavailability (RU) or new task arrival necessitates reallocating remaining tasks in a predetermined assignment plan. In this paper, we assume that if the executing time of tasks is controllable, it is critical to utilize the compressing ability of tasks to absorb the interference effects caused by design change. Consequently, a multi-objective reallocating approach with compressing executing time strategy (CETS) is developed to respond to design change so that the limited capacity of design resources can be utilized to the utmost extent. Moreover, it could reduce the reallocating cost and keep the stability of the system. We firstly design an advanced genetic algorithm for the optimal resource-task assignment with the objective for minimizing the maximum completing time. We assume that if a RU occurs, CETS can be carried out to solve this problem effectively as a reactive method.
1.1 Literature
1.1.1 Task Allocating/Reallocating
Two directions of task allocating have been considered in the existing literature. One is the mathematical modeling for allocating optimization [4], [6], [7], [35], [41], the other one is the algorithm optimization [5], [16], [32], [33], [34], [36].
The allocating problem of design task has some differences from manufacture scheduling problem, but the processing way of allocating/reallocating is consistent. Therefore, the method of manufacture scheduling problem can be an excellent reference to deal with the interference effects in CPD process. Tiwari et al. [52] provide references on models with conflict between time and quality in scheduling, and allow time saving to improve quality. Joglekar and Ford [27] construct the optimal resource allocation model to improve product collaborative design efficiency. Lapègue et al. [35] present an industrial problem which aimed at building employee timetables covering the demand given by a set of fixed tasks. Moreover, they refer to this problem as the shift-design personnel task scheduling problem with an equity criterion, in reference to the shift minimization personnel task scheduling problem. Jimenez et al. [26] present a scheduling model of radar design tasks to achieve both simple design and good performance. It is concluded that the allocating process consists of three stages in which the whole scheduling is divided into task prioritization, allocating algorithm, and temporal planning.
Most of the approaches for the problem of static allocating are often impractical because in the real-world design process, there are unexpected real-time events to disrupt this process. Therefore, dynamic scheduling is more realistic to deal with real-time disruptions.
The multi-objective task scheduling of large-complex equipment design is considered by Castro et al. [10], and a continuous time representation method based on uniform time grids is proposed. Mixed-integer nonlinear/linear programs are used to solve the short-term task scheduling problem and periodic task dynamic programming, respectively. Manufacturing firms often think about how to plan the production of a new order while containing the impact on the existing plan. Some researchers propose an alternative approach combining the most effective aspects of online scheduling and rescheduling. Dynamic scheduling can be divided into completely reactive scheduling, predictive-reactive scheduling, and robust pro-active scheduling. When solving the dynamic scheduling problem, principles of heuristics, meta-heuristics, multi-agent systems, and other artificial intelligence techniques are usually discussed.
1.1.2 Design Change Management
CPD is in a highly uncertain environment due to design change. Changes such as customer’s demand, temporary changes of design content, technological innovation, and so on could lead to the failure of collaborative resources and the decline of matching ability between resources and tasks. More seriously, they will make an unpredictable impact on the quality of complex product by deteriorating the stability and efficiency of the whole system. Managing the disruptions caused by design change is important for organizations to carry out CPD process since it fosters design process improvement and optimization. In the case study of Rolls-Royce, five-step uncertainty management method is used to analyze, quantify, and transfer the uncertain factors in complex product R&D in order to promote and optimize the design process, which has high practical industrial significance [49]. Johnson [28] proposes a framework for analyzing transformations of demand. Such transformations frequently stem from changes in the dispersion of consumers’ valuations which lead to rotations of the demand curve. Jarratt et al. [24] try to improve and optimize the product design process by analyzing the change trap of change operation in CPD. Wright [57] point out that design change based on driving force is not necessarily an obstacle. It should be regarded as an opportunity to promote the product quality and enterprise’s competitiveness. They provide a review of key publications by industrial and academic researchers relating to the management of engineering change. Matsushima et al. [42] mention that it is important to determine the type of change and resource commitment that is in need of change to achieve the modification of existing product design and to promote the perceived value of products. Furthermore, some literature propose that the analysis of propagation path and propagation impact of design change in process of product design can effectively prevent and avoid some risks caused by design change [2], [3], [11], [12], [17], [23], [25], [37], [63]. Reallocating is an effective method to respond to design change in the existing literature [1], [15], [19], [20], [29], [38], [51], [56], [59], [60].
1.1.3 Controllable Executing Time
In the current study, the strategies of coping with different types of interference are considered [30], [46], and the shortcomings of static scheduling and researches of dynamic scheduling in recent years are reviewed by Ouelhadj and Petrovic [45]. In the classical deterministic scheduling problems, most researches are carried out under the assumption that task executing time is a constant parameter. However, tasks allocating is developed by the decision makers in CPD process. Therefore, it is easy to understand that the executing time is flexible and learnable; i.e. the executing time is controllable by allocating a continuous or discrete resource to the executing sequences.
In order to minimize the effects of possible disruptions on a schedule, inserting idle time in the early scheduling to accommodate disruptions is a commonly used strategy. In any idle time insertion approach, when the executing time is constant and not subject to any adjustment, inserting idle time is an effective way to deal with dynamic uncertain allocating. However, in case of controllable executing time, when the resource time capacity is limited and fully utilized, inserting idle time requires applying extra compressing time. This increases the design costs. Moreover, the inserted time could be insignificant and wasted if no change occurs or it occurs after the time buffers. Vickson [53] is one of the first researchers to study scheduling problem with controllable processing time; he presents simple methods for solving two single machine sequencing problems when job processing time is a decision variable, and they have their own associated linearly varying costs. These are the problems of minimizing the total processing cost plus either the average flow cost or the maximum tardiness cost. After 1980, following the impetus of Vickson’s paper, sequencing problems with controllable processing time have been extensively studied by researchers [48]. When there is an urgent disruption caused by design change, we can adjust the executing time to respond to this problem. In various real-life design systems, the executing time is controllable. In such systems, an alternative rescheduling approach to inserting idle time is reaction to design change by resetting the executing time. Each processing time is a decision variable to be determined by the scheduler who can take advantage of this flexibility to improve system performance. Consequently, the limited capacity of design resources can be utilized more effectively.
1.2 Contribution
To the best of our knowledge, generating an assignment plan for allocating environments with controllable executing time, which can respond to disruptions with less reallocating costs, has not been studied well in the literature yet. Our work is the first attempt to employ reactive allocating with controllable executing time.
This paper contributes to CPD design research with a method named CETS for responding to interference impacts caused by design change. The motivation of this study originates from the needs of designers and engineers involved in CPD process and the practice that we can control the executing time in real-world design process. Executing time controllability provides us the flexibility in reallocating against unexpected disruptions by allowing changes on the executing time of the tasks.
The method proposed in this paper is able to respond to design change and support designers and engineers to optimize their design process. Our method can reduce the reallocating costs and simultaneously keep the system stability. The strategy and algorithm shall generate long-term competitive benefits in organizations arising from design process improvements.
The second research contribution presented in this paper is an empirical case study performed to demonstrate the method’s applicability in real CPD design environments. Also, the impact factor is discussed that may decide the applicable conditions of the proposed method. The priority of compression is remarkable. This motivates us to study more complicated disruptions in spite of the challenges.
Our case studies show that the proposed approach can achieve the effectiveness in the trade-off of change-adaptation cost and stability variation.
1.3 Organization
The remainder of this paper is organized as follows. In Section 2, the design change categories are discussed firstly, then we define the considered allocating environment and formulate the reallocating problem with a multi-objective. In Section 3, the proposed reallocating algorithm is introduced. We firstly use the adaptive multi-objective hybrid genetic algorithm and tabu search (AMOGATS) to establish the original allocation, then give the CETS algorithm for the time-cost minimization problem. In Section 4, the experimental design and results are discussed, and we give the conclusion and future work in Section 5.
2 The Multi-objective Task Reallocating
In practical design systems, numerous real-time events will occur during the executing process. The capacity of controlling design change determines the design efficiency, product performance and the response speed of market. Responding to design change is a central issue in the field of product design.
2.1 Classification of Design Change
The reasons for design change come from a variety of aspects, such as change of requirements, design repair, technological innovation and re-design. In the following section, the classification of design change is discussed in detail.
There are different criteria and perspectives on the classification of design change in the existing literature. Four key features are used to classify the design change by the Boeing Company. The classification is based on the scope and nature of change in literature [39]. However, initiated change and urgent change are classified by reference [13]. Design resource and design task are the most basic elements in CPD; adjustment and conversion of executing state of the design task and resource is the main reason leading to reallocating. Therefore, in this paper, we classify design change into two categories: Resource-related and task related [54], [55].
Resource-related concerns are the following: resource joint, resource withdrawal, unavailability, designer absence, tool failures, delay in arrival or shortage of materials, decrease of resource capacity, etc. Tasks cannot be performed by resource within a defined or random time interval when the resource is urgently unavailable.
Task-related concerns are the following: task joint, task cancellation, change on due date, task adjustment, early or late arrival of tasks, rush tasks, change on task priority or executing time, etc. Rush tasks may lead to the shortage of design resource relatively; change on due date may affect the expected completing time, and the adjustment will lead to a delay in resource preparation time, etc.
We assume that the time expression form in task-related change can be converted to the relative design resource, i.e. the final presentation of design change is the unavailability of resource relatively. Similar to machine failure in production scheduling, RU is most frequently considered in design task reallocating. RU occurs randomly and repairs after a period of time. In this paper, we select RU as representative design change.
2.2 Reallocating Method
The original allocation, adaptive delay right-shift (RS) and completely reallocating (CR) are discussed to respond to design change. CR is the conventional method in reactive scheduling. However, CPD process involves the coordination of different domains with different organizations. CR will lead to a huge information exchange between tasks and resources and a high management costs so that it is difficult to make an effective control. When the negative impact of cost is greater than the optimization of time, the response is regarded as an infeasible program. In order to keep the trade-off between time and cost, CETS is proposed to respond to design change. Case studies are conducted by comparing the system performance between CR and CETS. Moreover, we apply CR and CETS to respond to design change in different combinations of change factors. When design change occurs, we can choose the strategy according to the proposed approach. As mentioned in most literature, the performance of reallocation is subject to the arrival time, duration, and time interval of RU. It will be tested and verified by our computational study in Section 4. More details about CETS are presented in the following section.
2.3 Initial Virtual Design Unit-Task Allocating
Task allocating decision should be coordinated carefully to achieve the most efficient system performance. The required granularity of design resource is related to the division granularity of design task. Accordingly, tasks cannot be completed with one unitary resource. In this paper, we make an effective integration of design resource monomer named Virtual Design Unit (VDU). A single designer, computing device, software/hardware, tool or design knowledge model, which cannot complete a certain design task individually, is defined as a design resource monomer. To realize effective utilization of design resource, VDU [9] is taken as the basic task executing unit. According to design activity types, VDUs are divided into design scheme demonstration, geometric modeling, grid generating, programming, mathematic modeling, engineering analysis, simulation, experiment, test, scheme check, and prototyping. Single design resource is the element of VDU. According to design task requirement, VDUs are dynamically generated by organizing different design resources. VDUs are connected by design task sequence in design process. Design tasks are assigned with short time, low cost and equilibrium load; designers in VDU are the executing subject.
The task allocating is subject to the following technological constraints and assumptions: (1) Each VDU can perform only one design sequence of any task at a time. (2) A sequence of a task can be performed by only one VDU at a time. (3) All VDUs are available at time 0. (4) Once a sequence has been processed on a VDU, it must not be interrupted except when VDU is unavailable. If a sequence is interrupted, the remaining executing time is equal to the total executing time minus the completed executing time. (5) A sequence of a task cannot be performed until its preceding sequences were accomplished. (6) The capacity of compression can be calculated by experience, namely, the lower bound and upper bound of executing time are known in advance. (7) The number of applicable VDUs and the capacity of each VDU is known.
A general definition of task allocating problems with controllable task executing time may be stated as follows: product design is decomposed into n independent tasks after analysis, T∈{T1, T2, …, Tn}, i=1, 2, …, n; they are to be executed on m VDUs, U∈{U1, U2, …, Um}, k=1, 2, …, m. Design tasks can be decomposed into a series of determined design sequences: {Tij, j=1, 2, …, Ni}; Tijk is the executing sequence of task Tij on VDU k for j=1, 2, …, Ni, where Ni denotes the total number of sequences in each design task. TijU is the set of VDUs which can execute the sequence Tij, i.e. the design sequence Tij can be completed by any VDU in TijU.
The ultimate goal of task allocation is to assign design tasks to VDUs optimally with a certain design sequence and determine the starting time and completing time to ensure that it can be completed in the delivery period. Basic definitions and descriptions are shown as below.
Definition 1:
Xijk is the discrimination condition that the design sequence Tij is executed by the VDU k. Yijpqk is the priority discrimination condition that the design sequence Tij and Tpq are executed by the VDU k.
Definition 2: In order to simplify the problem, some hypotheses are given as follows. The relationship between design tasks is unambiguous after their decomposition; the design resources they need are explicit and limited during dispatching period. The task arrival time and standard execution time is known in advance through the analysis of historical data and experience, and the sub-tasks cannot be interrupted during its execution process. Each VDU can only execute at most one design sequence at the same time, and each design sequence can only be completed by one VDU. The number of design tasks is more than or equal to VDUs. The relationship between the sub-tasks assigned to each VDU is non-preemptive. The following equalities and inequalities are used to express the constraints based on the needs of a model.
Sijk, Eijk, and Cijk represent the starting time, executing time, and completing time, respectively. Formula (1) guarantees that VDU k can execute the follow-up sequence only after the completion of the front task. Constraint (2) denotes the propriety of sequence in each task; Tij starts after the consummation of Ti(j−1). Formula (3) is the executing time constraint, and (4) denotes that Tij can only choose the executing unit from TijU. The completing time constraint is described in formulations (5) and (6).
2.4 Compressing Executing Time Strategy
Responding to design change with immutable executing time has been studied well in the current literature. However, the biggest difference between design task reallocating and job shop scheduling is that the designer as the executing subject is self-adaptive to the reality and more flexible as a certain design resource. In such system, the executing time can be shortened by the way of design innovation, efficient cooperation, additional money, overtime and so on. This feature determines that executing time of design task is variable and controllable, i.e. the executing time of a design task can be compressed by a non-linear consumption of resource.
2.4.1 Compression Range
In this paper, the Affected Operations Rescheduling in job shop scheduling [21] is referred to to decide the tasks set to be compressed. Tasks to be reallocated are judged based on the principle of feasibility and optimality, which include original allocated tasks being executed and tasks directly or indirectly affected by design change. Design tasks in completed state are not taken into account because they do not influence the results anymore. According to the extent of impact, we need to define the range of the least affected reallocating tasks and the priority to be compressed. Matching ordering strategy is adopted to judge the affected tasks from the tasks set in executing and waiting state. The illustrated example is shown in Figures 1 and 2.
An Initial Allocation.
Diagram of Task Allocating Conflict.
We assume that resource R2 is unavailable at the moment NOW. It leads to a direct delay of T41, T42. Moreover, it affects the follow-up tasks T43, T52, and T53. Thus, {T41, T42, T43, T52, T53} is the set of reallocation tasks. Then formula (11) in the Section 2.4.2 is used to define the priority of compression to match up with the original allocation. If the compression of {T41, T42, T43} achieves the consistency of match-up time and completing time of the last reallocating task, the remaining tasks maintain the original allocation.
2.4.2 Compression Measures
In CETS, tasks to be reallocated, compressed amount, and compression cost are essential to be taken into account during the compressing process.
Change-Adaptation Cost
Eijk can be compressed by a consumption cost of resource with a non-linear growth. In this study, we assume that
where a≥b>0, h>0, D presents the time of resource availability. Rik denotes the number of kth VDU carring out the ith group design task. Cik denotes single executing cost when kth VDU carries out the ith group of design task, Cik→(i+1)k denotes the design activity cost when kth VDU transfers from ith group of design task to i+1th type design task. nez and cez denote the number and hour cost of the zth kind of equipment. nms and cms denote the number and hour cost of the sth kind of man. ntt and ctt denote the number and hour cost of the tth kind of auxiliary tools. caux denotes the sum of other affixed cost. Lik denotes the hourly workload of kth VDU when kth VDU executes ith group of design task. For more details about the calculation of the formula (8) refer to [44].
Compression order(Oijk):f(y) is a convex function, so that the cost of compressing a period of time through multiple design sequences is significantly lower than the cost that a single design sequence compresses the same period of time. It is consistent with the actual situation. We first generate the optimal compressibility
We use the compound sequencing rule to determine the compression order, which is formulated as
f(y) is an increasing and convex function [18]. By solving the convex programming function, we can get the optimal compression
2.5 Objective Function
CR may result in instability and lack of continuity in practical allocations. This would increase the extra design costs attributable to what has been termed system nervousness [45]. Therefore, in this paper, we attempt to keep the balance between the allocation stability and efficiency. In existing literature, a multi-objective mathematical modeling is applied to construct the allocation/reallocation. The allocation efficiency and stability are considered in evaluating the solutions. The efficiency is measured by makespan (Cmax). The smaller Cmax implies higher resource utilization [22]. The most frequent way to measure the deviation of their task completing time between the original and the realized allocations is to compare the stability [43], [58]. It should be noted that the consumption costs of resources need to be considered during the reallocating.
In this paper, CETS is proposed by considering the optimal coordination of partially affected tasks to utilize the design resources more efficiently to optimize the reallocation of system. However, compression could lead to the change of design costs. The change-adaptation cost
Task allocation in CPD process is to assign the tasks to VDUs with definite orders based on the optimization goal of time and cost. The change-adaptation cost
The objective functions can be described as:
where STB denotes stability; CAC denotes change-adaptation cost; Cijk and
where α, β, and γ represent the weight of each objective, respectively; the value is discussed in Section 4.
3 AMOGATS Algorithm for Reallocation
Task allocation of CPD is a multi-objective combinatorial optimization problem, i.e. a NP-hard problem. Currently, such problems have been well solved by using heuristic algorithms, but the solving ability and the scale of adaptation have various limitations. As for a multi-objective optimization problem, hybrid intelligent optimization algorithms which combine heuristic algorithm with other intelligent optimization techniques present superiority [60]. Therefore, the AMOGATS is developed by combining tabu search, genetic algorithm, and self-adaptation. We introduce the unique memory function of tabu search (TS) into the evolutionary search process of genitic algorithm (GA) and construct a new crossover tabu search recombinantion (TSR) operator. To improve the climbing ability of GA, we treat TS as the mutation operator of GA by the strong climbing ability of TS algorithm, tabu search mutation (TSM) operator is used to represent it. AMOGATS integrates the advantage of multi-starting points of GA and the strong memory function and mountain climbing ability of TS. Meanwhile, it introduces the self-adaptive idea to shorten the process of population evolution.
The general process of the proposed approach is summarized in Figure 3.
Flowchart of the Reallocation in Response to Design Change.
The steps of the proposed AMOGATS are as follows:
Step 0: Parameter setting. The maximum iteration Ngen, population size Npop, crossover probability pc, and mutation probability pm are given in the implementing stage.
Step 1: Initialization. We initialize evolution generation t=0, the number of allocations n=0, ts,n=0, the starting time Sijk=ts,n, where ts,n denotes the nth starting time.
Step 2: Chromosome encoding. It generates the initial population P(popsize). The case of task allocation is coded and decoded in the way described in [8], [31], [50], where each chromosome is a design task assignment scheme. The chromosome consists of six gene fragments, which are sequences that are assigned to each VDU. Different design tasks contain a different number of design sequences and the corresponding relationship of the chromosomes. The coding of design tasks and the VDU is shown in Figure 4.
Step 3: Condition updating. According to the requirement, we first select out of the matching VDU and update the design capability attribute of design unit. The VDU with shortest executing time is prioritized under the premise that there is no conflict.
Step 4: The executing time of the first design sequence Ti1 of each task can be calculated according to formula (6). Otherwise, the start time of Ti(j+1) equals the completing time of task Tij. And then, the completing time of j+1th design sequence can be calculated plus the executing time. After obtaining the executing time of all the design sequence, we can acquire the fitness value of the design task according to formula (18).
Step 5: Select the chromosome with the number of Npop from the population by a roller method, then place Npop chromosome into a mating pool. The probability of the individual being selected can be formulated as
Step 6: Crossover. The crossover randomly generates the number ri, range [0, 1], i=1, 2, …, Npop. If n<pc, the ith chromosome in the mating pool serves as a crossed parent; it produces Npop parent chromosome;
Step 7: Mutation. The mutation randomly generates the number ri, range [0, 1], i=1, 2, …, Npop. If n<pm, we schedule the TSM to perform the mutation operation for the ith chromosome in the mating pool.
The pseudocode for the operation of the restructuring operator TSR is shown as below.
begin
if fitness of x>average value of population, then accept x;
else
if offspring x is not in tabu list
accept x
else
choose the better of two parents to the next generations;
end
The pseudocode for the operation of the mutation operator TSM is shown as below.
begin
t=0; set the best solution x0=x; set T;
while termination condition not atisfied, do
t=t+1
move x to x′
update (x; x0; tabu list)
end
The self-adaptive function of crossover probability pc is defined as
(19)The self-adaptive function of mutation probability pm is defined as
(20)where fmax denotes the maximum fitness value, favg denotes the average, f is the larger fitness of the two individuals, and f* is the fitness value of who is to be mutated.
Step 8: If t<Ngen, the Zth(1≤Z≤Npop) the chromosome needs to be taken out from the population to Step 3. If t=T, output Fi, and the scheduling scheme, go to Step 10.
Step 9: Let t=t+1; the next generation P(popsize)t+1 can be obtained after selection, crossover, and mutation operations of P(popsize); go to step 8 with a loop implement.
Step 10: In order to respond to design change, whether the reallocation point is triggered and which strategy we use are judged before implementing. If CETS is chosen to respond to design change, the zone of influence needs to calculated. And then, the set of tasks to be reallocated needs to be selected out to implement the compressing to make the scheme run in accordance with the initial one as soon as possible. The value of parameter value {αi}=(α1, α2, α3, α4) has a greater impact on the reallocation, so we need to choose the appropriate value {αi} to reduce the compression cost and ensure stability. Here we use the basic genetic algorithm to achieve the coding, individual evaluation, and other genetic manipulation. Then the optimal parameter value {αi*}=(α1*, α2*, α3*, α4*) can be output in the evolutionary process. The compression sequence of design sequence can be determined by the estimation time and the duration of the design change.
An Example of Chromosome.
The duration is composed of the sum of approximate duration and the time that had been performed with change disturbance. We use four-dimensional nonnegative real number vector {αi}=(α1, α2, α3, α4) as the individual population, and dPIS/dNIS is the fitness value of the giving parameter (α1, α2, α3, α4), where dPIS and dNIS denote the Euclidean distance of the two-dimensional vector
where AT is the compression task zone, and Lmin is the minimum match-up time. In this paper, Lmin is equal to the completing time of the last task in the compression zone. The completing time of the last task before change occurs is expressed as W1. W2 is the affected time of change disturbance, which is equal to the sum of approximate duration and the time that the design task has been performed with change disturbance.
If we choose CR, when design change occurs; reallocating must be launched immediately. All the executing sequences must be canceled at that moment, and put all remaining tasks into the set to be executed. Then the system will implement the CR to formulate a new allocation. Since CR is not our focal point, in this paper we do not make a detailed description.
Step 11. This step mainly determines the compression length of each design sequence according to the trade-off of the stability, change-adaptation cost objectives, and the related constraints so that the new program table can be consistent with the original program table. When the next change arrives, the reallocation is required again; go to Step 10.
4 Computational Study
4.1 Experimental Design
To evaluate the performance of proposed method and algorithm, a case study is conducted in this section. In the computational study, MATLAB7.0 is used to write the simulation and test program according to the contract terms and actual progress.
Literature [61] chooses a wind turbine as a complex product; they consider a wind turbine that consists of thousands of parts. Generally, a wind turbine contains 10 key components, which are equivalent to 10 design tasks. Suppose that each task includes three processes. All tasks are carried out by three designers from different institutions. However, as the designer is the executing subject in this paper, we use VDU to perform the task so that some modification on the suitability and the attributes of VDU is needed. An aircraft is a typical complex product; therefore, we use one multi-task allocating in aircraft design process to illustrate our organizing and allocating process [9]. In literature [9], 17 design tasks and six VDUs are established; the attributes of design tasks are known in advance. Moreover, capacities of VDUs who are suitable to execute the 17 design tasks are involved in this literature. In literature [9] every design task consists of nearly 20 design sequences. If we consider the disruptions caused by design change, it is difficult to match up the timetable based on our existing experimental conditions.
Therefore, in order to be much closer to reality and convenient for further discussion, we design data that could mirror the real-world design environments. The data we use are designed by weakening the complexity of data from literature [9], but utilizing the capacity description of VDU. Simultaneously, the setting that every design task consists of different sequences is much more real than the setting in literature [61] that every design task has the same design sequence. Moreover, this requires the comparison of the performance of three algorithms and three strategies.
Thus, we design modified experimental data to mirror the aircraft design process, which can also be used in our further study. CPD is an applicable work pattern. An aircraft as a complex product consists of four VDUs and 12 design tasks, and each task has the corresponding design sequence. Due to the different composition of VDUs, the competent capacities of VDUs are different for different design tasks.
The essential information of the executing time, competent executing sequence, delivery period, and activity type is shown in Tables 1 and 2, where [225, 235] means that the optimal duration of the due date is from 225 to 235. Without loss of generality, we let 30 be the number of chromosome population, 100 be the evolutionary generation, crossover ratio parameters Pc1=0.8, Pc2=0.6, and mutation ratio parameters Pm1=0.1, Pm2=0.01 [61]. After developing the static initial design scheme, we consider a design change that U1 is unavailable at time 100 and recovers at the 130. We use the target weight value α:β:γ=5:1:1; this value is better than any other value according to the literature [47]. In the formula of change-adaptation cost, coefficient h is randomly obtained from [1, 2.8], a/b is randomly defined as an interval [1.1, 2.9], and uijk is generated from Eijk×Uniform [0.5, 0.9] [14].
Task Attributes.
| Design Task Group | Due Date/h | Executing Sequence |
|---|---|---|
| Geometric modeling T1 | [225, 235] | [T11, T12, T13, T14 ] |
| Meshing T2 | [125, 130] | [T21, T22, T23] |
| Aerodynamic analysis T3 | [255, 265] | [T31, T32, T33, T34, T35] |
| Meshing T4 | [265, 275] | [T41, T42] |
| Geometric modeling T5 | [105, 115] | [T51, T52] |
| Structural strength analysis T6 | [85, 95] | [T61, T62] |
| Geometric modeling T7 | [95, 105] | [T71, T22] |
| Structural strength analysis T8 | [220, 210] | [T81, T82, T83, T84] |
| Flight dynamics analysis T9 | [260, 270] | [T91, T92, T93, T94] |
| Structural strength analysis T10 | [230, 235] | [T10,1] |
| Trajectory simulation T11 | [230, 240] | [T11,1, T11,2, T11,3] |
| Flight dynamics analysis T12 | [260, 270] | [T12,1, T12,2, T12,3, T12,4] |
Design Capability Attributes of VDUs.
| U1 | U2 | U3 | U4 | ||||
|---|---|---|---|---|---|---|---|
| Task | Time | Task | Time | Task | Time | Task | Time |
| T11 | [19, 20] | T13 | [26, 28] | T14 | [23, 24] | T12 | [25, 26] |
| T12 | [23, 24] | T21 | [30, 32] | T22 | [35, 36] | T22 | [41, 42] |
| T31 | [25, 27] | T32 | [31, 32] | T23 | [35, 36] | T31 | [19, 20] |
| T33 | [24, 26] | T34 | [20, 22] | T33 | [29, 30] | T34 | [26, 28] |
| T41 | [22, 23] | T42 | [15, 16] | T34 | [22, 24] | T41 | [17, 18] |
| T51 | [26, 28] | T51 | [30, 31] | T35 | [20, 22] | T52 | [30, 32] |
| T62 | [28, 30] | T52 | [36, 37] | T41 | [34, 36] | T61 | [21, 22] |
| T83 | [25, 26] | T62 | [38, 40] | T61 | [17, 18] | T72 | [22, 24] |
| T84 | [31, 32] | T71 | [21, 22] | T62 | [34, 36] | T81 | [23, 24] |
| T92 | [27, 28] | T82 | [25, 26] | T71 | [20, 22] | T91 | [32, 34] |
| T94 | [25, 16] | T83 | [29, 30] | T81 | [20, 22] | T92 | [25, 26] |
| T11,1 | [21, 22] | T91 | [38, 39] | T84 | [25, 26] | T10,1 | [18, 20] |
| T11,3 | [20, 22] | T93 | [21, 22] | T91 | [38, 40] | T11,3 | [23, 24] |
| T12,4 | [23, 24] | T11,2 | [34, 35] | T93 | [23, 24] | T12,2 | [22, 24] |
| T12,1 | [34, 36] | T10,1 | [18, 20] | T12,3 | [28, 30] | ||
| T12,3 | [28, 30] | T11,2 | [26, 28] | T12,4 | [25, 26] | ||
4.2 Computational Results
The Gantt chart of static allocation is shown in Figure 5; when a design change occurs, the reallocation considering RS shown in Figures 6–8 are the Gantt chart of CR and CETS, respectively. The performance of each strategy is shown in Table 3.
Gantt Chart of Static Allocation.
Gantt Chart of RS.
Gantt Chart of CR.
Gantt Chart of CETS.
Performance of Strategy.
| RS | CR | CETS | |
|---|---|---|---|
| Cmax | 266 | 250 | 238 |
| STB | 342 | 304 | 69 |
| CAC | 450 | 762 | 425 |
| f | 2122 | 2316 | 1684 |
From Table 3, the Cmax in static scheme is 238. After responding to the change, Cmax in RS changes to 266, and CR changes to 250, but it keeps 238 in CETS. In terms of stability, we have used RS which only allocates the affected operations and preserves the stability of the resource allocation. When we use CETS, the design sequences are prioritized by the sequence T33, T14, T12, T34, T11,3, T12,4, T84, T93, the compression times corresponded to are 9, 7, 6, 6, 5, 5, 5, 4, 4, where 1 is the minimum time unit. Thus, the performance of CETS is better than CR, and CR is better than RS. In CAC, RS is better than the CR in keeping with reality, because there is no change in matching situation in RS, but CR has changed the matching situation to increase the change-adaption cost. However, the CAC in CETS is much lower because the increase in compressing cost has been counteracted by the decline of design cost. The overall performance shows that CETS is significantly predominant than CR and RS. In addition, the management cost of CETS should be much lower than CR and RS, although it is not taken into account in this paper. Therefore, a conclusion can be drawn based upon the aforementioned results that when responding to change, consumption of CETS is the lowest, and the performance is the best in the simple simulation theoretically.
To verify the superiority of the algorithm, AMOGATS is compared with the adaptive multi-objective flexible dynamic scheduling algorithm (AMOGATS) in [40], and the hybrid tabu search genetic algorithm (GATS) in [62]. After 20 times independently running, the curve of the function value is shown in Figure 9; the algorithm performance is shown in Table 4. The population size is 100, and the maximum evolutionary number is 100. In GATS, the crossover rate is 0.8, and the mutation rate is 0.1. Considering fairness, in AMOGATS and AMOGATS, crossover ratio parameters are set as Pc1=0.8, Pc2=0.6, and mutation ratio parameters are Pm1=0.1, Pm2=0.01 [61].
Results of AMOGATS, AMOFDSA, and GATS.
Performance Comparison of Algorithm.
| Algorithm | f̅ | f̅1 | f̅2 | f̅3 | t̅ |
|---|---|---|---|---|---|
| AMOGATS | 1684 | 238 | 69 | 425 | 15.265 |
| AMOFDSA | 3360 | 244 | 250 | 1810 | 19.623 |
| GATS | 3520 | 250 | 260 | 2010 | 24.265 |
Based on the result shown in Figure 9, GATS appears convergence after 40 generations. AMOFDSA is in the steady search process from 7 to 17, and the optimal solution is obtained in 22 generations. AMOGATS is in the steady search process from 7 to 10, and the optimal solution emerged in 14 generations. Also, in the same operating environment, the running time by GATS is 24.265, AMOFDSA is 19.623, and it is 15.262 by AMOGATS. Thus, AMOGATS has a higher performance and shorter running time than the others.
4.3 Further Discussion
In order to choose the strategy rationally, the applicable condition of CETS and CR is discussed in this section. We consider the change of performance with different uncertain factors, such as arrival time, recovery time, and interval of RU. In this study, we assume RU arrival at three intervals [80–100], [40–60], and [10–20]; randomly, the recovery time is in the interval of [10–30], and the interval of RU is from range [10–20]. There are three phases of arrival times, and we use numbers 1, 2, and 3 to stand for [80–100], [40–60], and [10–20], respectively; the three levels of recovery time are (10, 2, 30), and three intervals of RU are (10, 15, 20). Two strategies (CETS, CR) are considered. Therefore, a total of 3×3×3×2=54 experiments are conducted.
The results of performance comparison are shown in Table 5.
Impact of Change Factor on Reallocation Performance.
| AT | RT | Int. | Stra. | f̅ | f̅1 | f̅2 | f̅3 | Dev. | t̅ |
|---|---|---|---|---|---|---|---|---|---|
| 1 | 10 | 20 | CETS | 1530 | 238 | 104 | 236 | 2.14% | 6.41 |
| CR | 2249 | 240 | 296 | 753 | 2.35% | 6.53 | |||
| 15 | CETS | 1866 | 238 | 112 | 564 | 2.86% | 6.62 | ||
| CR | 2372 | 242 | 356 | 806 | 3.21% | 6.85 | |||
| 10 | CETS | 2033 | 238 | 146 | 697 | 3.54% | 6.94 | ||
| CR | 2448 | 242 | 384 | 854 | 4.17% | 7.15 | |||
| 20 | 20 | CETS | 2084 | 238 | 156 | 738 | 4.39% | 6.59 | |
| CR | 2548 | 246 | 432 | 886 | 5.34% | 6.95 | |||
| 15 | CETS | 2222 | 238 | 163 | 869 | 6.56% | 6.86 | ||
| CR | 2622 | 246 | 478 | 914 | 7.22% | 7.32 | |||
| 10 | CETS | 2319 | 238 | 175 | 954 | 6.07% | 7.21 | ||
| CR | 2746 | 246 | 512 | 1004 | 8.53% | 7.53 | |||
| 30 | 20 | CETS | 2370 | 238 | 182 | 998 | 4.34% | 7.46 | |
| CR | 2920 | 248 | 584 | 1096 | 4.20% | 7.78 | |||
| 15 | CETS | 2429 | 238 | 196 | 1043 | 5.12% | 7.85 | ||
| CR | 3002 | 250 | 625 | 1127 | 5.34% | 7.98 | |||
| 10 | CETS | 2500 | 238 | 214 | 1096 | 7.16% | 7.88 | ||
| CR | 3111 | 250 | 702 | 1159 | 7.36% | 8.24 | |||
| 2 | 10 | 20 | CETS | 2773 | 269 | 286 | 1142 | 5.71% | 8.13 |
| CR | 3332 | 271 | 714 | 1263 | 5.92% | 8.37 | |||
| 15 | CETS | 2861 | 269 | 306 | 1210 | 6.13% | 8.37 | ||
| CR | 3481 | 274 | 745 | 1366 | 6.25% | 8.58 | |||
| 10 | CETS | 3069 | 269 | 398 | 1326 | 6.41% | 8.54 | ||
| CR | 3726 | 276 | 783 | 1563 | 7.22% | 9.21 | |||
| 20 | 20 | CETS | 3213 | 269 | 436 | 1432 | 5.86% | 8.64 | |
| CR | 3850 | 279 | 852 | 1603 | 6.91% | 8.74 | |||
| 15 | CETS | 3377 | 269 | 496 | 1536 | 4.35% | 8.85 | ||
| CR | 3996 | 282 | 904 | 1682 | 4.62% | 8.97 | |||
| 10 | CETS | 3612 | 269 | 634 | 1633 | 3.79% | 9.54 | ||
| CR | 4356 | 296 | 1154 | 1722 | 3.95% | 9.68 | |||
| 30 | 20 | CETS | 3957 | 269 | 826 | 1786 | 4.25% | 8.78 | |
| CR | 4577 | 301 | 1236 | 1836 | 4.66% | 9.23 | |||
| 15 | CETS | 4233 | 269 | 1026 | 1862 | 4.85% | 9.16 | ||
| CR | 4795 | 304 | 1311 | 1964 | 5.35% | 9.35 | |||
| 10 | CETS | 4703 | 269 | 1342 | 2016 | 3.25% | 9.67 | ||
| CR | 5062 | 312 | 1360 | 2142 | 5.46% | 9.73 | |||
| 3 | 10 | 20 | CETS | 5374 | 343 | 1496 | 2094 | 5.21% | 9.53 |
| CR | 5238 | 323 | 1420 | 2163 | 7.87% | 9.65 | |||
| 15 | CETS | 5489 | 343 | 1569 | 2203 | 4.35% | 9.88 | ||
| CR | 5453 | 335 | 1480 | 2205 | 4.85% | 9.97 | |||
| 10 | CETS | 5936 | 343 | 1976 | 2298 | 8.97% | 10.34 | ||
| CR | 5651 | 341 | 1536 | 2245 | 9.63% | 10.34 | |||
| 20 | 20 | CETS | 6038 | 343 | 2013 | 2410 | 4.76% | 10.46 | |
| CR | 5898 | 358 | 1596 | 2310 | 5.35% | 10.58 | |||
| 15 | CETS | 6251 | 343 | 2136 | 2512 | 5.48% | 10.74 | ||
| CR | 6028 | 352 | 1682 | 2400 | 6.26% | 11.13 | |||
| 10 | CETS | 6383 | 343 | 2245 | 2586 | 8.34% | 11.26 | ||
| CR | 6139 | 356 | 1725 | 2423 | 9.16% | 11.68 | |||
| 30 | 20 | CETS | 6664 | 343 | 2453 | 2634 | 3.88% | 11.54 | |
| CR | 6623 | 359 | 1812 | 2496 | 4.45% | 11.57 | |||
| 15 | CETS | 6943 | 343 | 2695 | 3016 | 6.36% | 11.76 | ||
| CR | 6898 | 362 | 1936 | 2533 | 6.57% | 11.86 | |||
| 10 | CETS | 7114 | 343 | 2786 | 3152 | 7.56% | 11.98 | ||
| CR | 6459 | 366 | 2016 | 2613 | 7.63% | 12.11 |
AT, arrival time; RT, recovery time; Int., interval of RU; Stra., CR and CETS strategies; Dev., deviation; t̅, running time of algorithm.
When RU scale is consistent, the influence on allocation is positively associated with recovery time and arrival interval. Similarly, when recovery time is given, the effect is positively associated with arrival interval as well. The more forward the RU occurs, the greater the impact is. The general trend is consistent in other conditions. But there is a special time point when the arrival time is 3, recovery time is 10, and interval is 20. In this point, CR is better than CETS in overall performance, but Cmax is still lower than CETS. When complexity increases, the responding capacity of CETS decreases gradually. On the contrary, CR keeps a growth trend in comprehensive index in response to more complicated problems. The ambiguity in the description of coordination cost and the different measurements are the main reason. However, in real design process, there are differences when the practitioners choose design objective. Some of them concentrate on the design cost, some concentrate on others. If practitioners only concentrate on the time, they can choose the CETS. Otherwise, the balance between change-adaptation cost, stability, and tardiness penalty should be considered according to the actual situation.
5 Conclusions and Future Directions
In this paper, CETS for task reallocating in CPD process is developed and compared with the performance of CR and RS. The results obtained from an extensive experiment show that CETS outperforms CR and RS in completing time, change-adaptation cost, and stability. It is worth mentioning that, in this paper, the stability is measured by the completing deviation from the original allocation. If the change of design sequence is taken into account, the stability of CR could be much worse than we measured. This shows the superiority of CETS in turn. The applicable conditions and related factors of CETS are analyzed. When the problem is more complicated, an irregular pattern appears on the performance. The advantage of CR shows gradually. Furthermore, computational examples show a high efficiency of AMOGATS algorithm to solve the reallocating problem.
In this work, the response to design change is based on the existing initial scheme. However, the probability allocations of resource unavailability, recovery time, and arrival of new tasks have been investigated in some recent studies. Thus, the priority of compression can absorb design change automatically if combined with probability distributions to formulate a robust predictive-reactive schedule. In this way, loads of manual input and adjustment can be eliminated from CPD process, which should be completed more precisely in future study.
Acknowledgments
The authors are grateful for the support of the National Natural Science Foundation of China (NSFC, Project No. 71571023) and the Chongqing Graduate Student Research Innovation Project (Project No. CYB15018).
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