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不确定情形下大型工程项目的多目标均衡优化
合伙人  徐武明   《管理科学与工程管理》   201111
Abstract: A large-scale construction project is an uncertain complex system with multiple objectives. Despite extensive efforts to effectively tackle this problem in recent research, the understanding of the objectives and uncertainty should be more accurate. Accordingly, after proposing hypothesis and definitions, which better fit the engineering project on reality, a double-uncertainty multi-objective tradeoff optimization model with randomness, fuzziness and five objectives - time, cost, quality, safety, and environmental friendliness is established in this paper. And then the evaluation function is formatted, combining with network planning technique, group decision-making technique, multi-objective decision theory and uncertainty theory, to solve the model with the help of genetic algorithm by Mat lab tools. At last, it’s described and verified by actual examples, with higher solution efficiency. Therefore, the results showed that the method is valid, has some theory value and practical meaning.
Keywords: Multi-objective Optimization, Uncertainty Analysis, Decision Making, Genetic Algorithm, Large-scale Construction Project,
1 Introduction
Since the large-scale construction project is a difficult task with multiple objectives[1] and complex environment and is big influence to society and economy, there is a great necessity to study the large-scale construction project both theoretically and practically.
The typical method in optimizing project construction is network planning technology[2], such as planning evaluation technology and critical path method. In the past researches, the optimizing objectives in project construction mainly considered the time, cost and quality [3-9]. In the recent studies, safety was noticed [10,11], but as far as authors know, no one considered that the importance of environmental friendliness to large-scale construction project[12]. The large-scale’s influence to environment is ruinous, any other objectives are not worthy of mention, thus the objective- environmental friendliness cannot be neglected. In the optimization of time, cost and quality, due to the uncertainty of construction objectives, random theory is the most used in early period, then fuzzy theory for a few studies[13-16].  However, some studies with fuzzy theory only discussed the construction time. Randomness and fuzziness are confused or disordered by some researchers using fuzzy theory[14], some hypothesis do not conform to the project reality in real-world projects. Therefore, the research results are not practical[15-16].
At first, more practical objectives are extensions of time, cost, quality, safety, and environmental friendliness in this paper based on the previous research. Secondly, the writer analyzes and differentiates that the uncertainty of five objectives is different. Then a double-uncertainty multi-objective tradeoff optimization model with randomness and fuzziness is established through uncertain network planning technology, using theory and method of multi-objective decision-making and group decision-making. Genetic algorithm is applied to satisfactory solution of decision-making proposal. So decision makers have more flexibility to obtain a satisfactory result. Finally, it is verified by an example. 
2 Hypothesis and Definitions
Hypothesis 1  In practice, the main objectives in a large-scale construction project are: time, cost, quality, safety, environmental friendliness. Since different influencing factors, each objective is of obvious uncertainty. 
Hypothesis 2  Among the uncertainty of these five objectives- time, cost, quality, safety and environmental friendliness, time and cost is random, but not fuzzy[1,17]; quality, safety and environmental friendliness is with double uncertainty, which is randomness and fuzziness.
Definition 1  As above-assumed, quality, safety and environmental friendliness is with fuzziness, shown as natural fuzzy language, and defined by triangle fuzzy numbers, as shown in Figure 1.
 
Fig. 1.  Triangle Fuzzy Numbers.
Hypothesis 3  A construction project comprises of many works (procedures), there are many constructing modes (methods) for each work, and the time for each constructing mode may be different. For cost, quality, safety and environmental friendliness, it is similar, each mode may be different.
This hypothesis, comparing with infinite hypothesis of working mode, more conforms to the reality[5,15].
As above-mentioned, time and cost of each constructing mode for each work (procedure) is random variable, its quality, safety and environmental friendliness is random fuzzy variable.
Hypothesis 4  the time of each constructing mode for each work is β- distribution, the expectation of the time needed is:  , Variance is :  .[18]
Where, , ,   is respectively, under normal condition, minimum estimation, most possible estimation, and longest estimation for finishing the work by this constructing mode. The estimation can be set according to the experience and quota by inviting the experts. The normal condition here indicates the extreme impossible factors are excluded. The following mentioned is similar.
Hypothesis 5  The time distribution of each work is individually separated. The critical path is decided by the expected value of working time needed. The expectation of project total time (E(T)) is sum of the expected value of working time needed for each work of critical path. The variance (VAR (T)) is sum of each variance for each work of critical path. When there is too many works, the finishing time for each work is not in an absolute position, according to central limit theory, the total time obeys the normal distribution N(E(T), ). [18]
Hypothesis 6  The total cost of project includes both direct and indirect cost, where, the direct cost of project is sum of the direct cost for all works, the direct cost is, among normal time, in inverse proportion with time, the indirect cost is proportional to the total constructing time.
Hypothesis 7  The quality of each work is individually separated, the working quality is proportional to time under normal condition. The project quality is the weighted sum of working quality.
Hypothesis 8  The safety of each work is individually separated, the working safety is proportional to time under normal condition. That is: the time is longer, work is safer. The project safety is the weighted sum of working safety.
Hypothesis 9  The environmental friendliness of each work is individually separated, the working environmental safety is proportional to time under normal condition. That is: the time is longer, the environment is friendlier. The project environmental friendliness is the weighted sum of working environmental friendliness.
Definition 2   Direct cost change rate of work i:  = , quality change rate:  = , safety change rate:  = , environmental friendliness change rate:  = .  Where  = minimum estimation and  =maximum estimation for finishing the work i according to the constructing mode under normal condition;  = the direct cost of related  ,  = the direct cost of related  ;  = the fuzzy quality of related  ,  =the fuzzy quality of related  ;  = the fuzzy safety of related  ,  =the fuzzy safety of related  ; = the environmental friendliness of related  , = the environmental friendliness of related  .
3 Multi-objective Tradeoff Optimization Model
T = target time of large scale construction project, C = target cost,   = target quality,  =target safety,  = target environmental friendliness. The project can be divided into n works. The work i (i=1,……,n) has  constructing modes;  (≥0 random variable)is the working time of work i constructed according to the chosen mode,  ; (≥0)is the initial time of work i;  is project indirect cost, where,  is the fixed part, k is the indirect cost of unit interval; ( , ,≥0)is respectively the weight of quality(safety, environmental friendliness) for work i.
Multi-objective tradeoff optimization for construction project expects short time, low cost, good quality, sound safety and most friendly environment. Thus, the Multi-objective tradeoff optimization model for construction project is established as follows.
                                  (1)
        s.t   - - ≥0     j is the followed work of i.
            ≥ ≥ ≥0
Through α- cut of fuzzy sets, α∈ [0, 1], and the multi-objective tradeoff optimization model for construction project is changed to Format 2:
                                  (2)
        s.t   - - ≥0     j is the followed work of i.
            ≥ ≥ ≥0
4 Model Solution
The model is the decision-making problem with fuzzy and random double uncertainty. For quick solving the model, target   will be evaluated by E , the expectations of   which are been as random variables. Please see hypothesis 5. E  can also be written as , other variables are same. 
Format 1  can be shifted to Format 2  for solution with fixed α(α∈[0,1]). For certain α,superior solution  and inferior solution   will be figured out.                                                                      (3)
The model is multi-objective, using ‎Linear weighted sum method, the weighted target is( , , , , ), its sum is 1. The evaluation function is  , the multi-objective decision-making problem is transferred to the optimization problem of evaluation function .
Therefore, each target is standardized, the solution of model(2)can be transferred to (3)with superior solution: 
 =  ( , , , , )    (4)
With inferior solution:
 =   ( , , , , ) (5)
 , , , , , , , in right end of Format(4)~ (5) is the solution for single target function without taking other targets into consideration. The T which locates in the upper-right corner the brackets is transposition. 
The large-scale construction project includes excessive works; each work also includes many constructing modes. Since genetic algorithm can browse the best solution and second-best solution all over the space, the continuous space is not required, which is suitable for solve the network planning optimization of many modes for different work[19], then the genetic algorithm is chosen. The study subject is individual of the group, the omosome structure is that each omosome comprises of n gene, which represents n works of construction project. The gene position represents working code, the gene value is for working mode.       
The probability of the individual to be ed will be higher with roulette ion operator when its fitness value is bigger. With the strategy of remain the best and second best individual, the best and second best individual will not be cross-calculated. Cross operator is uniform cross. Mutation operator is random integer variation within gene value. The fitness value is the  , of which evolves in the increasing direction. 
5 Case Study
A large-scale construction project: S---hydro-electric power station, with a sub-project, that working relationship is as Table 1
Table 1 Working Relationship
Working code 1 2 3 4 5 6 7 8 9
Followed work 2,3 4,5 5,6 7,8 8 8 9 9 
N0.of mode 4 2 4 4 4 4 4 2 4
Fixed indirect cost   is 810, 000 RMB, the indirect cost of unit interval k is 120000 RMB/day.
     Through group decision-making method and inviting three experienced experts, Weight , ,  is respectively given for quality (safety, environmental friendliness) of work i. The arithmetic mean value is as Table 2
Table 2  Working Weight
Work 1 2 3 4 5 6 7 8 9
 
0.11 0.08 0.09 0.1 0.07 0.06 0.08 0.11 0.1
 
0.1 0.08 0.09 0.9 0.06 0.06 0.08 0.12 0.1
 
0.1 0.09 0.09 0.1 0.07 0.07 0.07 0.11 0.1
And the target weight ( , , , , ) will be respectively given by three experts, its arithmetic mean value of target weight is (0.2,0.28,0.2,0.16,0.16).
Basic parameter of work 1 for each mode is as Table 3, the unit of unit interval indirect cost is ten thousand RMB Yuan. Work 2-9 is omitted because of layout.
Table 3  Basic Parameter of Work 1 for Each Mode 
Work code Exp-
erts Work-
ing mode Finishing time(day) Direct cost Quality Safety Environmental Friendliness
    
 
 
 
 
 
 
 
 
 
 
1 1 1 10 12 15 165 110 ave gd usf sf ave fri
  2 11 13 15 160 112 ave gd ave sf ave fri
  3 12 15 18 150 120 ave vgd ave vsf ave vfr
  4 15 17 19 150 120 gd vgd sf vsf fri vfr
 2 1 10 12 14 170 110 pr gd usf sf ufr fri
  2 12 13 15 160 110 ave gd ave sf ave fri
  3 12 15 17 155 120 gd vgd ave sf ave vfr
  4 15 17 20 150 120 gd vgd sf vsf fri vfr
 3 1 11 13 15 165 105 ave gd usf sf ave fri
  2 10 13 15 160 110 pr gd ave sf fri fri
  3 12 15 18 150 110 ave vgd ave vsf ave vfr
  4 15 18 20 155 120 gd vgd sf vsf fri vfr
(ave=average, gd=good, vgd=very good, pr=poor, sf=safe, vsf=very safe,
usf=unsafe, fri=friendly, vfr=very friendly, ufr=unfriendly,)
αof α- cut set is 0.5, its superior solution is solved first, that is   corresponding to  .
The time expectation of the mode of each work can be solved according to Hypothesis 5. Every single objective value was calculate by network planning technique:  =189.2,  =12565,  =0.932, =0.921, =0.913。
The cross operator of genetic algorithm is 0.8; mutation operator is 0.01, group numbers are 20, and realized by Matlab. The space of this case is , but terminated at 30 generations, only browsing 600 solutions in average, thus the solving efficiency is relatively high. The superior solution of most satisfactory 0.5-cut set is:(225.8,13175,0.903,0.885,0.871), the most satisfactory working mode group is:(3,2,2,3,2,1,4,1,2), related evaluation function is  =0.935.
For inferior solution of 0.5-cut set, the solution is similar, then any α- cut set can be solved.
6 Conclusion
Then, the following conclusions were obtained in this paper.
1) Five-objectives including project time, cost, quality, safety and environmental friendliness have been considered in this paper. Multi-objective tradeoff optimization model is established with full scale consideration, better meeting the large scale construction practice and society need, multi-objective tradeoff optimization model is established.  
2) Objective uncertainty caused by project complex has been considered, double uncertainty for random and fuzziness is obviously divided, and uncertain multi-objective tradeoff optimization model is established.
3) The group decision-making theory is adopted for the preference of different decision-maker.
4) Formatting evaluation function, to solve the conflict of multi-objectives, combining with the theory and method of network planning technology and uncertainty (random and fuzziness), the model is effectively solved.
5) The genetic algorithm is applied for solving the model by Mat lab tools, with a higher efficiency.
The above-described method is more suitable for large scale construction project, and has some theory value and practical meaning.
The further research is mainly model solution and application to engineering practice.
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