Development and application of a hybrid genetic algorithm for resource optimization and management. O. O. UGWU* & J. H. M. TAHâ . *Department of Civil ...
Engineering, Construction and Architectural Management 2002 9 4, 304–317
Development and application of a hybrid genetic algorithm for resource optimization and management O. O. UGWU* & J. H. M. TAH† *Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China, and †Division of Civil Engineering and Construction Management, South Bank University, London, UK
Abstract Resource selection/optimization problems are often characterized by two related problems: numerical function and combinatorial optimization. Although techniques ranging from classical mathematical programming to knowledge-based expert systems (KBESs) have been applied to solve the function optimization problem, there still exists the need for improved solution techniques in solving the combinatorial optimization. This paper reports an exploratory work that investigates the integration of genetic algorithms (GAs) with organizational databases to solve the combinatorial problem in resource optimization and management. The solution strategy involved using two levels of knowledge (declarative and procedural) to address the problems of numerical function, and combinatorial optimization of resources. The research shows that GAs can be effectively integrated into the evolving decision support
INTRODUCTION Many decisions in construction projects usually involve assigning resources from one task to another. Such decisions are often required at various levels of a project life cycle: conceptual level – when the project manager is concerned with the total cost and project feasibility, tender appraisal; submission level – when contractors are concerned with preparing reasonable and economic cost estimates that has to be matched with project resource requirements; and at the operational level – when site and contract managers have to deal with the realities of daily operational decision-making. For a given project the resources assigned determines the method(s) of construction. Therefore, the decision problems often demand evaluating the best way to distribute available resources over different tasks that are necessary for a successful and efficient completion of the project. Such resource assignment and optimization problems demand efficient combinatorial computations if all possible options are to be considered, and decision-making facilitated. This is true irrespect-
systems (DSSs) for resource optimization and management, and that integrating a hybrid GA that incorporates resource economic and productivity factors, would facilitate the development of a more robust DSS. This helps to overcome the major limitations of current optimization techniques such as linear programming and monolithic techniques such as the KBES. The results also highlighted that GA exhibits the chaotic characteristics that are often observed in other complex non-linear dynamic systems. The empirical results are discussed, and some recommendations given on how to achieve improved results in adapting GAs for decision support in the architecture, engineering and construction (AEC) sector. Keywords combinatorial optimization, decision support systems, distributed project management, genetic algorithms, resource optimization
ive of the project management level. Consequently, research into efficient methods of resource optimization has always been an area of interesting study on its own. Previous research works on resource optimization investigated the use of deterministic models in construction decision-making, while other works investigated the use of stochastic models in solving the problem (Paulson et al., 1987; AbouRizk & Shi, 1994; Smith et al., 1995). However, despite the long search for simulation models that will receive acceptance by practising engineers, deterministic models still remain the preferred method for studying planning and scheduling in the construction industry. Some authors (Schexnayder, 1997) have argued that deterministic models come very close to the daily practise of engineers and are therefore favoured. Such models enable practising engineers to harness their experiences when studying and verifying the effects of physical features of resources. Consequently, there still remains the need to investigate simulation models that takes the physical characteristics of resources into consideration. Genetic algorithms
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(GAs) offer potential solutions in developing such stochastic-simulation models. This is because such physical characteristics can be encoded as a set of parameters that determine the final cost of the resource components, using the features of a GA. The objective of this study is to investigate the potential application of GA systems in the general resource selection problem domain. The purpose is to establish how such efficient computational techniques can be applied to facilitate decision-making in the area of resource optimization and management within the framework of a distributed decision support environment. The focus is on project management at various levels of strategic decision-making, and ensuring that any choice of optimal strategy is underpinned by a careful analysis of the benefits and costs associated with implementing all the possible alternatives. In addition, the sensitivity of the output to changes in certain parameters that GAs need during execution will be examined. The aims of this paper are as follows: •
•
•
•
to give a comprehensive treatise on GA and their potential applications in the context of resource optimization and management; to demonstrate the application of combinatorial design techniques (i.e. the interaction between mathematical modelling and computing technology such as databases) in solving complex multidimensional problems; to discuss an empirical investigation on the reliability of the proposed GA system as a decision processing component in resource management; and to highlight the challenges and problems in the development and deployment of GA and other evolutionary techniques for decision support. It is intended to provoke some serious research questions in construction information technology research.
BACKGROUND Genetic algorithms belong to the family of artificial intelligence techniques that are increasingly being employed to solve optimization problems. Such algorithms mimic the operations of natural selection when searching for optimal solutions. The power of their use in applications is derived from their ability to combine numerical parameter optimization with combinatorial searches within an application domain. Genetic algorithms are therefore uniquely suitable for solving multidimensional optimization problems
such as resource selection in construction. In practice, the application of a GA involves designing artificial chromosome structures that represent a simple genetic model of the computation, and then implementing the genetic operators by simple bitmanipulation operations. Problem-domain analysis and encoding constitute substantial activities in designing a GA as a solution to an optimization problem. The basic building blocks that influence the efficiency and performance of a GA is the schemata from which the genetic model representation is constructed. The underlying details of the schemata theory are discussed in the seminal book by John Holland (Holland, 1975). Genetic algorithms have been successfully applied in different optimization problems including the famous Travelling Salesman Problem (TSP) in Operations Research. Some of the applications in design and construction management problems include: oil pipeline network optimization (Goldberg, 1989), structural optimization for truss roofs (Koumosis & Georgiou, 1994, Nagendra et al., 1996), determination of the laying sequence for a continuos girder reinforced concrete floor system (Natsuaki et al., 1995) and resource scheduling (Chan et al., 1996). Bennet et al. (2000) report on the development of a GA-based decision support system (DSS) for location of new major housing allocations. Rafiq et al. (2001) discuss the use of structured GA (SGA) to generate and evaluate different feasible design solutions concurrently, within a DSS framework. Borkowski & Grabska (2001) discuss the application of graphs in layout optimization and highlight the potential applications of GA in the layout optimization of trusses. Griffiths & Miles (2001) present a research project that is investigating the application of improved GA to optimize shape discovery in design, using two-dimensional string representations for the genetic search. Soibelman & Pen˜a-Mora (2001) describe a distributed multireasoning mechanism that incorporates GA and case-based reasoning within a multiagent system environment to provide designers with design solutions using a set of user-defined parameters and constraints. The problem domain is the conceptual phase of the structural design of tall buildings. The above catalogue of research projects shows an increasing interest on the development and application of GA-based systems. Majority of the research works on GA still focuses on encoding and representing the required solution as fixed length character strings (chromosome structure) and this same
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approach was adopted in our study. However, while most of the GA applications still focus on onedimensional string processing, the work reported in this paper extended the basic GA by investigating its application in a two-dimensional matrix problem. In general, most GAs employ three primary genetic operators: Reproduction, Crossover, and Mutation. Details of these genetic operators are exhaustively discussed in the referenced GA textbooks (Holland, 1975; Goldberg, 1989; Davis, 1991; Rawlings, 1991), and such finite details do not fall within the scope of present discussion. The functionality of a GA is maximized if it is used in an unconstrained optimization problem. However, GAs are usually applied to constrained optimization problems (COPs), by assigning penalty functions that transform them into unconstrained problems (Goldberg, 1989). The major disadvantage of this approach is that because the penalty functions are often arbitrarily assigned, some constraints may be violated by good solutions that are close to the border of feasible region. Other approaches include using ‘genetic repair’ to cushion the effects of possible constraint violations by the resulting GA solutions (Paredis, 1993). These approaches enforce some ‘synthatic correctness’ and consequently, they may not be acceptable in optimization problems that demand both numerical function and combinatorial optimization – a distinct characteristic of resource selection problems. Some authors and researchers have advocated that other tools such as greedy algorithms and constraint programs be incorporated to sift through an optimization problem before GA is finally called to solve the function and combinatorial aspects of the problem (Davis, 1991; Rawlings, 1991; Watson, 1995). This argument also supports the systematic approach to planning, and the authors agree with the thinking. The work that is reported in this paper represents a significant advancement in GA applications by integrating with project databases. In this study, the Structured Query Language (SQL) processing algorithm is used to sift through candidate resources in the database that satisfy duration constraints before the combinatorial optimization. This approach improves distributed resource optimization and project management. The ensuing sections describe the problem domain, as well as the solution strategy we adopted. The coding system that was found suitable for the problem and the experimental design for studying the system behaviour are also discussed. Finally, the empirical results obtained after analysing the output data are presented and recommendations given for further research.
THE NATURE OF RESOURCE ASSIGNMENT PROBLEMS Resource allocation as a network problem In order to investigate the suitability of GA for resource assignment and optimization, it is necessary to examine the problem in a generic context. In this regard resource assignment is viewed as analogous to network problem. The following sections discuss the characteristics of resource-assignment problems when viewed in this context. Fig. 1 is a graphical illustration of a resource selection/ combinatorial problem, showing the tasks and resources network. The problem description is outlined below: •
•
Each ellipse in the network (X1–Xm) is a node that represents a construction task, and each circle (Y1– Yn) represents different resources to be assigned. Thus, the problem depicted here is how to assign resources (Y1–Yn) among tasks (X1–Xm) when it is possible to use the resources (or a combination of resources) in completing any of the tasks, subject to constraints on allowable combinations. The following characteristics of the problem can be deduced from the network shown in Fig. 1 (Ugwu & Tah, 1998): – It is a function-evaluation/combinatorial problem. The optimization problem is to find the best traversal path in the network that minimizes the total cost of the project tasks. – In order to generate a solution space, the GA traverses through the network to create a new chromosome. This chromosome (bit string) results from a ‘permutation’ of a list of the cost indices that is encoded in the problem space and then generated by scrambling through the order of the nodes. This is a coevolutionary process. – The problem is epistatic – solutions over the feasible region are closely coupled and small
Figure 1 A network of the resource-assignment problem.
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•
alterations in the nodal weightings trigger cumulative effects (perturbations) in the solution space. Consequently, this affects the fitness of each traversal path, and also has some impact on the efficiency of a GA system, often leading to termination at local optima (Paredis, 1993, 1995). Each ellipse has multiple cost values, resulting from its interactions with the circles. These interactions can be translated into a payoff matrix of order (m · n) by using the generalized resource-allocation formulae described in the following section.
Mathematical model formulation The resource-based mathematical model expresses the total cost of a construction process as a function of the respective tasks performed using available resources, and the corresponding resource productivity and economic attributes (e.g. resource unit costs). The model considers resource optimization and management as a generic problem, and is built upon two fundamental sets of class objects: •
•
a project that consists of at least one task and the task(s) at hand required to be completed as part of the project – this task completion is a transformation process; and the resources that are required to execute the above project task(s).
The mathematical model is given by Equations (1) and (2) below (Winston, 1994): X Minimize rt ðxt Þðt ¼ 1; T Þ ð1Þ X ð2Þ subject to: gt ðxt Þ W where W is the units of a resource available, T the number of activities to which the resource can be allocated, gt(xt) the units of resource that are used by activity t, and rt(xt) the associated cost of using the resource gt(xt). A structured modelling approach was employed to translate this network into the corresponding payoff matrix (Geoffrion, 1987, 1988, 1992). This is shown in Fig. 2. The resulting payoff matrix yields the grid given in Equation (3): aij rt ðxt Þði ¼ 1; m : j ¼ 1; nÞ
ð3Þ
where aij corresponds to the cost or duration values for a given locus i, j in the matrix table (Fig. 2). The application of the matrix is illustrated in the genetic state-space search (GSSS) shown in Tables 2–6 in the Case Study section.
Figure 2 Payoff matrix structure of the resource-assignment problem.
While certain data values such as the task quantity are defined by the user (based on project details) the output of a given resource is determined by certain productivity factors, such as size, capacity, correction factors – (resource attributes), and the type and nature of the material encountered at site – (material attributes). For a given resource within each resource group, there is therefore a corresponding cost and duration attached to its usage. The task and resource attributes are both indexed directly to their respective classes/ objects and stored in the project database. The model has been applied in the earthwork operations domain and details of the application domain are discussed in (Ugwu, 1999). Table 1 summarizes the variables expressed in the mathematical model: Constraints in evaluating the objective function Time constraints (task and project duration) A set of duration constraints must be satisfied for a given assigned resource. The general expression for the duration constraint is given as: X XQmt ði; j; NÞ Dt ð4Þ Rot ði; j; NÞ i j where Rot is the output of a given resource R in performing task t, Qmt is the quantity of task under consideration, and Dt is the task duration. The general objective function also allows for a mixture of resources. In generating the mathematical model, the unit costs of resources are assumed deterministic here and are user-specified. The unit costs of labour, and plant are also assumed constant in this study but it may vary
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Table 1 Description of variables in the mathematical model. Indices, sets and relation p* ˛ P* r˛R p˛P l˛L m˛M t ˛ T = {e.g. e, h, f, c} Constants Ur for r ˛ R Variables Po Lo Qmt for r ˛ R DP, DT Cr (for r ˛ R = {plant, labour, material}) Constraints DT £ DP C‡0 ru £ ra (for r ˛ R)
Objective function
Projects (P* – various types of construction projects) Resources (R contains three types of resources: plant, labour, material) Plant is an ordered set classified by functional and operational uses Labour is an ordered set classified by type Material is an ordered set, classified by types Task is an ordered set by sequence of construction (e.g. excavation, haulage, filling, compaction in earthworks construction) Unit cost of resource (assumed deterministic and is user specified) Plant productivity (but assumed constant over section i, j) Labour productivity (but assumed constant over section i, j) Quantity of tasks handled by a resource group over the section i,j as extracted from project contract drawings and specifications but is user defined Project and task duration, respectively – determined by specifications in the conditions of contract Cost of a given resource element r, the cumulative sum of which gives total cost of construction/earthwork operations over the section(s) of interest to the user Duration limit constraint Cost limit constraint over section i, j Resource utilization constraint: resource utilization ru must be less than or equal to resource availability ra. The user certifies satisfaction of initial availability constraints As given in Equation (1)
with work conditions at various sections along a project profile. Previous works have been undertaken to generate models and programs that compute and simulate these unit costs (Easa, 1987, 1988, 1989) and it is not considered to be within the scope of this research. Equation (1) is the objective function that underpins this research. Equation (4) was imposed by using a parameterized query to sift through all possible candidate resources in the project database, before the preprocessed data set is passed on to the GA for functional and combinatorial optimization. This approach is underpinned by the fact that the existing SQL processing algorithms are extremely powerful, robust, and very suitable for the kind of constraint manipulation that is desired in this type of combinatorial problem. The next section discusses the incorporation of this objective function in the formulated genetic model. MODEL FORMULATION The genetic model representation of the resource assignment problems incorporates two important decision-making parameters – cost and duration. Using such a model would ensure that a user’s choice of construction resource(s) or resource combination is underpinned by a careful analysis of the costs and
benefits associated with implementing all the possible alternatives. In addition, the sensitivity of the output to changes in certain of the parameters that GAs need during execution was examined as part of the experiment designed for the system testing/validation. The ensuing section describes the algorithmic procedures of the hybrid GA. A hybrid genetic algorithm for resource assignment This section describes the hybrid GA that is integrated with a project database to perform combinatorial optimization. The database maintains the following task-schedule information – task/activity ID, names and description, activity durations, assigned resources, resource IDs and the corresponding productivity and economic factors, etc. The project database was implemented in Microsoft Access and used for persistent storage of task-schedule data. The hybrid GA uses parameterized structured query that is passed to the database engine, to extract details of tasks and resource productivity/economic attributes from the database. The extracted data are then used to process the associated costs and duration for each resource assigned to various tasks. Figures 3 and 4 show the database and structural relationship between the
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Figure 3 A view of the database table objects.
Figure 4 Relationship between the database table objects (project, tasks, and resources).
various database table objects – project, tasks, resources and resource assignment. This level of data and information storage improves the robustness of the GA because the services it provides (functional and combinatorial optimization) is independent of the data on which it acts in performing such services. The distinct feature also means that the imposition of genetic operators such as reproduction, crossover, and mutation do not result in an
arbitrary loss of information, as the knowledge about the problem domain on which the combinatorial optimization takes place is stored in the project database. Figure 5 shows the formulated hybrid GA with all its delineating features. The algorithm modifies the simple GA (Holland, 1975; Goldberg, 1989), in order to incorporate the specific data structure requirements of the problem. It is based on generating and mapping a fitness network
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Figure 5 Hybrid genetic algorithm for resource selection and optimization.
that constitute the genetic search space. This fitness network which encapsulates information related to a given resource or construction method and tasks, was then mapped into a set of genes with the associated cost values within a defined chromosome structure (a oneto-one genetic mapping). Solutions include identifying optimum combination of resources and tasks that
minimize the total cost of construction over the feasible region. The initial constraint fitness maps the value of each gene as the cost of completing the tasks with a given resource. An initial population was generated from which the algorithm learns and imposes the genetic operators and consequently coevolves other feasible solutions within
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the search space. Therefore, the fitness network serves a dual purpose by integrating two general paradigms: genetic search and state-space search. Some authors have advocated the adoption of this GSSS approach in solving COPs (Paredis, 1993, 1995). This is the approach we adopted in solving the problem. However, a salient feature of this hybrid GA is that all analytical computations are based on the actual resource productivity and economic data extracted from the project database. Ugwu (1999) and Ugwu & Tah (1999) discuss details of the decision-support framework and the underlying system architecture that underpin this level of integration of the GA in a prototype DSS. The genetic model The proposed genetic model/coding encapsulates the payoff value of a gene at a given locus. The genetic search space is illustrated in the system validation section (see Tables 1–5). The string representation is given in Equation (5). j¼1;N Chromosome: Cij ; Dij i¼1;M ð5Þ where Cij is the cost value of a gene, Dij the corresponding duration, and i, j define the locus of a gene in the chromosome structure. In this model, a gene defines a type of resource (e.g. construction plant) and the locus defines the task to which the resource has been assigned. Thus, values of a gene at a given locus corresponds to the cost and duration of executing a given task with a particular assigned resource. For example, a 7-bit chromosome defined as 1011001 represents a resource allocation model that assigns two different resources to seven different tasks. This type of binary chromosome representation can precisely match to a given set of resource assignments provided the chromosome contains enough bit strings to define the various possible resource assignments. In this model, the GA maintains
a population with fixed size of chromosomes (length and depth). CASE STUDY A pipe laying project was selected for this case study because it involves earthwork operations which was initially chosen as a test bed for the prototype implementation of the proposed model. The example project used for the validation is based on a detailed typical construction project information as documented in Carvalbo & Turner (1969). The requirements for resource optimization and project management were identified and analysed from the project documents. The following tasks were identified: Huts delivery, Huts assembly, Workshop delivery, Workshop assembly, Tanker greasing pit, Access Road Section 1, Trench for Tunnel Section VI, Trench for Tunnel Section VII, Trench for Tunnel Section VIII, Trench for Tunnel Section IX, Stone Filling over Tunnel Section VI, Stone Filling over Tunnel Section VII, Stone Filling over Tunnel Section VIII, Stone Filling over Tunnel Section IX. Two resources (a digger crane and truckmounted vehicle that could be converted for digging) are available for use in executing the outlined tasks. The extracted task and resource attributes were used to populate the appropriate database table objects. The problem involves optimizing the tasks and resource assignments from the case study project. GA system testing and validation The test on the GA system was carried in the context of sequential decision-making problem outlined in the preceding section. In order to apply GA, the project is broken down into the above component tasks, and there are two items of interest (control variables): 15 tasks, and two candidate construction resources that satisfy availability and duration constraints. A
Table 2 A sample preprocessed cost data sets for combinatorial optimization. Resource ID
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
ML1 TV1
30 150
30 150
10 50
10 50
10 50
10 50
100 500
40 200
40 200
20 100
10 50
10 50
10 50
10 50
10 50
Table 3 A sample preprocessed duration data sets for combinatorial optimization. Resource ID
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
ML1 TV1
3 1
3 1
1 0.5
1 0.5
1 0.5
1 0.5
10 2
4 1
4 1
2 0.5
1 0.5
1 0.5
1 0.5
1 0.5
1 0.5
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two-dimensional binary code string representation was used to model the decision-making parameters that are of interest (i.e. the cost and duration values associated with a given assigned resource). Each population consists of the binary strings 1 or 0, and both the bitstring breadth and depth are fixed. The GA evaluates the fitness of a given population set (chromosome) using preprocessed cost and duration data sets extracted from the database (Tables 2 and 3). The combinatorial optimization process begins with a set of initial populations, which are randomly generated for subsequent use by the GA (Tables 4–6). The user inputs include the values of certain GA optimization parameters, such as the number of generations, the mutation rate, and the crossover site. There are no strict restrictions on the user’s choice of GA optimization parameters. However, from the trial tests, the values used for number of generations and mutation rates ranged from 200–2000 to 0.0–0.3%, respectively. The GA uses these to learn and dynamically generate other populations (a coevolutionary process), and then generates, as a final output, the
cumulative cost of completing the tasks with various possible combinations of resources (i.e. method(s) of construction). The data structure of each output (chromosome) was designed to generate five sets of coded results, and each stream of the solution represents a possible combination of resources, and the associated cost. This is desired because in a DSS the user makes the final choice on the basis of other practical considerations, such as some logistics related to the project management. A typical structure of such output data is illustrated below: Optimization parameters: No. of generations ¼ 800 Bit mutation rate ¼ 0.1 Crossover rate ¼ 1 Strings correspond to construction methods (alternative combination of resources) String no. 0 ¼ 0 0 1 1 0 0 0 0 0 1 1 0 1 1 0 String no. 1 ¼ 0 0 0 1 1 1 0 0 1 0 0 1 0 1 1
Table 4 Genetic coding/representation of the search space using bit vectors. Resource combination
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
String String String String String
0 1 1 0 1
0 1 0 0 0
0 1 1 1 1
0 1 1 0 1
0 1 1 0 1
0 1 1 0 0
0 1 0 1 1
0 1 0 1 0
0 1 1 0 0
0 1 0 0 0
0 1 0 1 0
0 1 0 0 0
0 1 0 1 1
0 1 1 1 1
0 1 1 0 0
0 1 2 3 4
Table 5 Randomly generated initial population of costs (Cij). Resource combination
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
String String String String String
30 150 150 30 150
30 150 30 30 30
10 50 10 50 50
10 50 50 10 50
10 50 50 10 50
10 50 50 10 10
100 500 100 500 500
40 200 40 200 40
40 200 200 40 40
20 100 20 20 100
10 50 10 50 10
10 50 50 10 10
10 50 10 50 50
10 50 50 50 50
10 50 50 10 10
0 1 2 3 4
Table 6 Randomly generated initial population of duration (Dij). Resource combination
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
String String String String String
3 1 1 3 1
3 1 3 3 3
1 0.5 1 0.5 0.5
1 0.5 1 1 0.5
1 0.5 0.5 1 0.5
1 0.5 0.5 1 1
10 2 1 2 2
4 1 4 1 4
4 1 1 4 4
2 0.5 0.5 0.5 1
1 0.5 1 0.5 1
1 0.5 0.5 1 1
1 0.5 0.5 0.5 0.5
1 0.5 0.5 1 0.5
1 0.5 0.5 0.5 0.5
0 1 2 3 4
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String no. 2 ¼ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 String no. 3 ¼ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 String no. 4 ¼ 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 Value of objective function for each string String String String String String
no. no. no. no. no.
0: ¼ £630 1: ¼ £750 2: ¼ £350 3: ¼ £390 4: ¼ £510
Thus for a given test run, the system is able to outline the cost implications of various strategies – combinations of tasks and resources (Fig. 6). In its present form, the decision-maker decodes and interprets the output from the system. Further research is required to enable the GA system decode the output by itself, and classify the resulting bit strings in terms of the coded variables, i.e. the task names and description of the construction method(s). The experimental design to study the behaviour of the GA is discussed in the following sections, together with some of the observations that were made during the various stochastic-simulation sessions. Experimental design The experiment to study the behaviour of the GA system was designed to measure a set of output data for a given test run. Various statistical indicators were then used to measure the reliability of the system as a search/ optimization tool for decision support, by comparing the output with the actual best solution of the problem. It was also necessary to measure the sensitivity of the system to changes in the optimization parameters – number of generations and mutation rate. The following data were recorded: • •
maximum (best) value minimum (worst) value
• •
average value, and execution time in seconds.
The average value measures the overall quality of the system output for a given test run, higher values representing improved solutions. The execution time is an indication of the computing resources consumed by the system. For a given set of parameters, the program was executed 40 times and results were recorded in all cases. The output data were analysed using a statistical and a spreadsheet package. These tools facilitated a study of certain statistics and various levels of data exploration, which in turn revealed some interesting characteristics of GA systems (see Table 7).
Table 7 Performance data of 40 test runs. No. of generations Statistic
100
200
400
800
Pmutation = 0.0 Avg. minimum cost (£) Standard deviation Maximum (£) Minimum (£)
437.00 559.00 452.00 449.00 97.93 81.013 88.27 89.67 590.00 590.00 590.00 590.00 350.00 350.00 350.00 350.00
Pmutation = 0.1 Average minimum cost (£) 413.00 457.00 408.00 432.00 Standard deviation 84.44 114.72 74.12 115.98 Maximum (£) 590.00 790.00 590.00 790.00 Minimum (£) 350.00 350.00 350.00 350.00 Pmutation = 0.2 Average minimum cost (£) 410.00 442.0 Standard deviation 84.27 104.94 Maximum (£) 630.00 750.00 Minimum (£) 350.00 350.00
430.00 122.20 990.00 350.00
422.00 108.77 750.00 350.00
Pmutation = 0.4 Average minimum cost (£) 424.00 436.00 443.00 418.00 Standard deviation 84.27 125.04 93.87 83.12 Maximum (£) 750.00 750.00 830.00 790.00 Minimum (£) 350.00 350.00 350.00 350.00
Figure 6 Graph showing typical minimum cost profiles for the various outputs.
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Figure 7 Graph of the GA output for 40 test runs.
RESULTS The graphs of ‘best, worst, and average best’ were analysed (Fig. 7). Because of some chaotic characteristics exhibited by the GA system, statistical analysis of the output focused on determining the level of replication of output for different input parameters. The range of values, frequency distribution, and the proportion of results that satisfy a certain range are the indices that can be used to determine the reliability of the system as a DSS component. The solutions for which the maximum output lie over the range 85–100% of the ‘best solution’ were also measured (Table 8; Fig. 8). From the results of the experiments (Tables 4–8; Figs 6–8), the following observations are made: 1. GA systems are very efficient tools for complex combinatorial searches over a highly multimodal parameter space. 2. A cross-impact analysis was carried to study the effect of changes in the values of the optimization parameters on the output generated by the system on the different trial runs. It was observed that keeping one of the optimization parameters constant and varying the value of the other did not result in a predictable output pattern (Fig. 6). Also, an ANOVA test did not indicate a significant Table 8 Proportion of output within the range 85–100% of best solution. No. of generations Pmutation
100
200
400
800
0.0 0.1 0.2 0.4
77.5 85.0 87.5 77.5
85.0 75.0 75.0 67.5
80.0 65.0 72.5 70.0
80.0 72.5 77.5 70.0
correlation between the output data sets for different values of the parameters (GA system variables). Such chaotic behaviour is usually characteristic of other complex dynamic systems. The random nature of the stochastic modelling/simulation and the cumulative impact of the genetic operators (crossover and mutation) may induce these perturbations. 3. In general, mutation appears to distort the performance of the system – increasing the computation time without necessarily improving performance (Fig. 8). The variation in the execution time with the number of generations and the probability of mutation shows a logarithmic relationship. The time generally increases with the optimization parameters. This translates to computer processing resources utilized (cost), and can be very significant for a GA used in industrial applications. 4. Although the GA system generates the cost profile for various options as an output, a DSS will enable the project manager to investigate other options so as to be well informed of the consequences of taking a particular action. Hence, the GA system will be optimized if it is used as a component of a DSS in a wider context, and integrating of GA with the project database allows for a wide range of applications in real-time or real-life situations (Ugwu et al., 1998). DISCUSSION AND ANALYSIS OF THE RESULTS This paper has reported a research project that investigated a new approach to resource optimization and management using hybrid GAs and a solution strategy that is based on the object-oriented paradigm. The study also investigated the efficiency and behaviour of a GA system as a DSS component for distributed
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Figure 8 Proportion of the output within the range of 85–100% of the best solution.
decision-making in project management. The paper discussed a genetic model that represents the problem space for construction method selection and resource optimization/management. The hybrid GA uses a resource cost model that expresses project costs and duration in terms of task details, the physical characteristics (technical and productivity attributes) of construction resources, and the economic data such as resource unit costs. By adopting this solution strategy, the authors proposed two levels of knowledge utilization in GA-based resource optimization and management: •
•
Declarative knowledge – in which project, task and resource details are stored as FACTS in a database. These are then coded as a set of cost parameters in the two-dimensional genetic model developed for the research investigation; and Procedural knowledge – in which resource combinations are modelled as a stochastic process expressed in terms of the genetic operators (crossover, mutation, and reproduction) within the multidimensional genetic model.
By utilizing knowledge at the above two levels, the GA system generates the cost profile for various options (combination of the assigned resources) as an output. The project manager or user is also able to investigate other options so as to be well informed of the consequences of taking a particular action. The results demonstrate that a hybrid GA system is a potential reusable component for resource-optimization problems in various types of construction projects. The application has the following limitations in its present form: •
The representation of decision-making parameters in the genetic model is limited to two attributes: cost and duration. However, there is scope to expand on, and increase the number of project evaluation parameters to include: project location
•
factors, economic forecasts of the project, project risk factors and other investment decision-making parameters. The GA output is used to populate the taskscheduling database table object manually, so that the project management system can interface with the GA results. There is a need to automate the update functions of the database so that the GA output can be used to automatically update the database. Further work could also investigate the possible application of eXtensible Mark-up Language (XML) to extract display and/or interpret GA output data in various formats based on user preferences in a web-enabled distributed project management environment.
CONCLUSION This paper discussed the application of a hybrid GA to resource optimization and management. It described the formulation of a genetic model that addresses the specific problems of combinatorial optimization in managing construction resources. A suitable twodimensional data structure for the problem was also investigated. The GA interacts with a database and extracts the detail project and resource attributes for use in quantitative computations and combinatorial optimization. This approach of integrating GA with project databases is quite novel and adopting GA in this manner makes it a true global optimization technique. The study focused on the efficiency and behaviour of GA system as component(s) of a DSS. Our approach has been to examine resource assignment and optimization as a generic problem. The following observations were made: •
GAs can be effectively integrated into the evolving DSSs for resource optimization and management, and in solving other engineering problems. The study demonstrates that with adequate design of
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data structures a GA system can be a reusable component for resource assignment problems in various types of construction projects. Integrating a hybrid GA that incorporates resource economic and productivity factors would enhance the development of more robust DSS. This helps to overcome the major limitations of current optimization techniques such as linear programming. The output do not necessarily constitute the overall best solution in all cases but nevertheless, the speed of computation has been demonstrably very impressive as this supersedes any attempt to evaluate similar combinatorial trials manually. This suggests that the main advantage of a GA is that it guarantees a good solution over a large complex search space within a short time. GA should therefore be applied only to appropriate problems. The basic assumption at this stage of model development and testing is that each selected resource in the GA satisfies resource availability requirements. Although resource availability constraint is verified or relaxed by the user, further work is needed to incorporate greedy algorithms that sifts the resources to ensure that this fundamental assumption is not violated by any resource(s) in the solutions generated by the GA.
The result of the system validation shows that GA systems are very efficient tools for complex combinatorial searches over a highly multimodal space. The tests also reveal that although GA systems are capable of converging at optimal solutions within a very short time, they also exhibit some chaotic characteristics such as a complete absence of the optimal solution in a generation of results. However, such chaotic behaviours are often observed in other complex nonlinear dynamic systems. In addition a GA must have enough search space to minimize degradation of its performance. The major contribution of this work is that it extends the simple GA model by: (a) designing a GA that processes two-dimensional string objective functions, and (b) integrates with organizational databases in solving the multidimensional problem of resource management. This means that resource productivity factors can be independently updated and maintained in the database and dynamically extracted for function and combinatorial optimization by the GA even in a distributed environment. With this functionality for analytical evaluations, the hybrid GA exploits the power of the genetic operators in search and optimization over a large search space. The interaction with the project database(s) ensures that the knowledge of the tasks and assigned resources from which the analytical
results are computed is distinctly separated from the coded information on the genetic model that is subjected to the actions of the genetic operators. Thus the resource optimization problem is addressed here in a generic context while available task and resource attributes are used to populate the project database. Furthermore, the GA results can be propagated to other database table objects for integration with project management systems such as MS Project (Ugwu, 1999). This enhances distributed collaborative project management. The study reported in this paper has shown that GAs have huge potential generic applications in resource optimization in construction engineering and management, and that it is best suited for large and complex search problems. Most of the current works concentrate on developing GAs that optimize onedimensional string functions. This has very limited application in solving real life multidimensional problems such as resource selection and optimization. It is therefore recommended that further work be directed towards developing improved data structures that would facilitate solutions to such problems. Further research is also required to address the chaotic behaviours of GAs, and investigate its applications for decision support in other areas of engineering design and management. This will be a major step towards realizing its industrial applications in the architecture, engineering and construction sector. ACKNOWLEDGEMENT This research was conducted as a PhD programme funded by the Faculty of the Built Environment, South Bank University London, SW8 2JZ, UK. The authors also wish to acknowledge the useful comments and suggestions from the anonymous reviewers of this paper. REFERENCES AbouRizk, S.M. & Shi, J. (1994) Automated constructionsimulation optimization. Journal of Construction Engineering and Management, 120, 374–385. Bennett, L.D., Mawdesley, M.J. & Ford, M.K. (2000) Investigating a genetic algorithm based decision support system for the location of new major housing allocations within the local process. In: Second International Conference on Decision Making in Civil and Urban Engineering, Lyon France, 20–22 November 2000, pp. 897–908. Borkowski, A. & Grabska, E. (2001) Graphs in layout optimisation. In: Artificial Intelligence in Construction and Structural Engineering. Proceedings of the 8th International Workshop of the European Group of Structural Engineering Applications of Artificial Intelligent (EG-SEA-AI). ISBN 1
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