Multiple Resource Constraint Time-Cost-Resource Optimization Using Genetic Algorithm

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1 First International Conference on Construction In Developing Countries (ICCIDC I) Advancing and Integrating Construction Education, Research & Practice August -,, Karachi,, Pakistan Multiple Resource Constraint Time-Cost-Resource Optimization Using Genetic Algorithm Habib Fathi Construction Engineering and Management Student, Iran University of Science and Technology, Tehran, Iran Abbas Afshar Professor of Civil Engineering Department, Iran University of Science and Technology, Tehran, Iran Abstract Simultaneous optimization of time, cost, and utilized resources in a construction project is vital. This paper presents a GA based model for deteration of the best combination of the time, cost, and resources in a multiple resource constraint problem. The proposed model considers both resource allocation and leveling simultaneously. Since the problem is assumed to be resource constraint, resource allocations modify the schedules based on multiple resource restrictions. Besides, the basic concept of resource leveling, imization of M x (X-moment of resource histogram) is used to imize resource fluctuation. In addition to M x, the paper uses M y (Y-moment of resource histogram) in resource leveling process because simultaneous application of them improves it to take into consideration the resource utilization period. The paper uses weighted sum method for handling multi-objective optimization problem. Performance of the model is illustrated using a simple example project. Keywords Time-Cost-Resource Optimization; Genetic Algorithm; Resource Constraint Scheduling; Resource leveling.. Introduction Critical path method (CPM) is the most common method used for construction project scheduling. It takes into account time and deteres critical activities to imize project total duration. Due to surmount different limitations of CPM, several techniques have been proposed to overcome these shortcogs. Most of these techniques, namely time-cost trade-off analysis, resource leveling, resource allocation, and resource constraint scheduling, however, deal with distinct sub problems and thus can only be applied to a project one after the other, rather simultaneously. So, the need for practical and automated simultaneous optimization of different aspects of projects has increased in recent years, especially as a result of the application of new and emerging construction contracting and project delivery methods. A mathematical technique for schedule optimization was presented by Karshenas and Haber () considering resource constraint, cost of time, and monthly cash flow limit. The other study by Li () formulated an optimization model considering investment allocation, resource supply, and weather impact on productivity. Hegazy () presented a GA based algorithm which considered resource leveling and allocation and imized the moment of the resource histogram around the horizontal axis (time). Overall

2 Schedule optimization with respect to time, cost, and resource constraints using a spreadsheet model was developed by Hegazy and Ersahin (). In addition, Kandil and El-Rayes () devised a multiobjective automated construction resource optimization system in order to optimize time, cost, and quality of a project. The main purpose of this paper is to provide a management support for construction planners to optimally select the best possible implementation method for each activity with respect to project s overall performance throughout the planning stage. The model employs GA to imize the weight summation of time, cost, and resources distribution indices as the objective function. The paper assumes each activity has different execution methods accompanied by different time, cost, and multiple resource utilization. Besides, each kind of resources is supposed to be limited and has a constant limitation during project total duration. The paper employs double moment concept which originally was proposed by Hegazy (). In this process, M y (Y-moment of resource histogram) takes into consideration the resource utilization period in addition to M that considers the resource fluctuation. x. Problem Definition and Model Formulation The problem is defined as a combination of resource leveling under multiple resource constraint and a time-cost optimization problem. In other words, the project manager may look at the problem as a multiobjective one which simultaneously considers the interrelation between time, cost, and resources distribution. Activity sequencing in project management involves identification and documentation of the logical relationships among scheduled activities. Based on the project schedule, several methods may be noated for implementation of an activity. Each implementation method encompasses specific time, cost, and resources utilization. For example, in earthwork allocation, different fleets can be used with different productivity accompanied by particular time and cost. So, the project manager must detere a definite execution method for each activity relating to overall efficiency of the project. Disregarding the resource utilization and its associated constraints, the total duration of a schedule, T i can be detered as: [] ES = Max{ ( EF Lag) } n p + [] EF n = ESn + Dn [] LFn = Min{ ( LS s Lag) } [] LS n = LFn Dn [] TF = LS ES ) or ( LF EF ) n ( n n n n i n =,,..., [] T = Max{ EF } n k Where n is the activity number; k is the total number of activities; ES n is the early start of activity n; EF n is the early finish of activity n; LS n is the late start of activity n; LF n is the late finish of activity n; TF n is the total float of activity n; p is the number of activities which are predecessor of activity n; s is the number of activities which are successor of activity n; and Lag is the lag time between activities which are related to each other by precedence relationship.

3 The initial schedule modification and resource allocation based on resource constraints must be performed with the aim of calculating the resource constraint schedule s total duration (T ). In resource allocation process (sometimes referred to as resource-constraint scheduling), a limited number of resources distributed among different project activities while keeping the unavoidable extension of the project to imum. In another word, imum extension in project duration is considered. To fulfill this purpose, project activities must be sorted based on early start time. The activity with lower early start time acquires higher priority. Then, the limited resources have to allocate to each activity based on its rank. If during the early start and early finish of an activity at least one day is found that a specific resource utilization exceeds its limitation, the early start of the activity must be shifted to satisfy the resource constraint. Selected shifting value for an activity has to bring about the satisfaction of limitation of all kind of resources existing in project scheduling. When the shifting value for an activity is detered, the shifting value for the activity in next rank must be calculated. The maximum amount of modified early finish is the total duration of schedule T with respect to multiple resource constraint. The proposed model considers combination of X and Y moments of resource histogram in resource leveling process. Harris in proposed the imum moment algorithm for resource leveling and proved that lower value of resource histogram s X-moment (X is the time axis) is more desirable. According to Hegazy (), it can be expressed as: R T [] M x = ( Re source i= j= Demand ) Re j source Demand j i where R is the total kinds of project resources; and T is the total working days of project. In a more recent work, Hegazy () proved that while the imum moment algorithm can be used to compare among histograms in terms of resource fluctuation, it does not take into consideration the resource utilization period. The latter is very important to imize, particularly for equipment resources that are shared among projects or rented from external sources. Fig. shows two resource histograms which have the same X-moment whereas their Y-moment is different. Since the first histogram has lower Y-moment, it is better than the second histogram from resource utilization period perspective. It must be noted that for better results a simple modification can be used to calculate the moment M y around a vertical axis that corresponds to the first day the resource is employed in the project. This modification is illustrated in Fig.. M x =. M y =. M x =. M y =. Fig.. Resource histogram and moment calculation

4 The Y-moment of resource histogram can be calculated as (Hegazy ): R T [] M y = [ ( Re source Demand j ) ( j.) ] i= j= where R is the total kinds of project resources; and T is the total working periods of the project. Therefore, total amount of moments is represented by: [] M = M x + M y i.. Objective Function Fig.. Vertical axis modification for M y calculation Overall evaluation of a specific schedule is necessary in the context of underlying objectives. The paper considers the time, cost, and X-moment plus Y-moment of resources utilization histograms as three objectives of the problem. Using weighted sum method, a set of objectives are scalarized into a single objective by pre-multiplying each objective by a user-supplied weight. The weight of an objective is usually chosen in proportion to the objective s relative importance in the problem. Setting up an appropriate weight vector also depends on the scaling of each objective. To normalize the objectives, this paper uses the modified adaptive weighted approach (MAWA) as: [] Z = W D D D D D max + W C C C C C max + W M M M M M max in which W D, WC, WM are the weights of time, cost, and total amount of X and Y moment respectively; D, C, M are the duration, cost, and total moment of a possible schedule respectively; D max, D, Cmax, C, M max, M are the maximal and imal value of duration, cost, and total moment in the current population of genetic algorithm respectively; and γ is a very small positive number in order to prevent dividing by zero in objective function, and also does not permit objective

5 function to become zero because the model uses the inverse of objective function for reproduction scale in GA. It must be noted that because the implementation option of each activity in a specific schedule is definite, total cost of a schedule is calculated by adding the cost of selected option for each activity of project.. Solution Finding Procedure To search for the best combination of implementation method of each activity in a specific project using genetic algorithm, five steps must be tracked. Fig. illustrates different stage of this process in a flowchart. This procedure finds the optimum combination of executing methods for project s activities which fulfills resource limitation constraint. The paper uses the inverse of objective function for reproduction scale of each chromosome. Besides, tournament selection is applied for selection operator. In this operator, tournaments are played between two solutions and the better solution is chosen and placed in the mating pool. Start Initialize Population Analyzing Initial Schedules without Resource Constraint Resource Allocation Evaluation and Assign Fitness Reproduction, Selection, Crossover, and Mutation No Teration Condition? Yes End Fig.. Solution searching process

6 . Example Apllication A small example project is used to demonstrate the stages and merits of the model presented in this paper. Table shows the required information about the sequencing of project activities and implementation options of each activity accompanied by their relative time, cost, and utilized resources. Activity Table. Activities, predecessors, and activity options of example project Predecessors ---,,,,, Implementation Method Duration (Days) Cost Resource Requirements Per Day R R R Daily Resource Limits The model was solved with a GA optimizer developed in Visual Basic by the authors. Population size of, with crossover and mutation probability of and percent were used respectively. Number of generation was limited to with W D =., W C =., and W M =. as the model parameters. Results of the solution of the model are shown in Table. The solution of model has duration, unit cost, and unit total histograms moment.

7 Table. Results of example project Activity Selected Option Modified ES With respect to selected options for implementation method of each activity, the total duration for initial schedule of project is working days. Therefore, in order to satisfy multiple resource limitation constraints, the total duration has to be increased by working days (from to days). This increment is the result of early start time modification... Fitness Value..... Function Evaluation Fig.. Convergence of the model Rate of convergence of the model to locate a near optimum solution is presented in Fig.. Although the searching process is extended to number of function evaluation which corresponds to generations, the near optimum solution remained more or less unchanged for the th generation (i.e. number of function evaluation). Fig. represents the rate of resource utilization for each resource type during the project implementation period. Inclusion of M x and M y in the model has resulted in well leveled resource utilization for resource type and. For resource type, With respect to Fig. (c), if the modification in vertical axis does not perform, the total amount of M y will be equal to unit. Whereas, if M y is calculated around the vertical axis that corresponds to the first day the resource is employed (day ), it will be unit and gives a better perspective of resource utilization period. It implies that the first activity does not use resource type.

8 Resurce Amount Day (a) Resource Resource Amount Day (b) Resource Resource Amount Day (c) Resource Fig.. Utilized Resource histograms of solution

9 . Summary and Conclusion This paper develops a multi-objective optimization model that selects the best combination of implementation option for each project activity with respect to time, cost, and total moment of resource histograms around horizontal and vertical axis. The proposed model considers both resource allocation and leveling. It takes into account M x to decrease resource fluctuation in resource histograms and M y to consider resource utilization period. The value of M y, as such, gets higher as the resource remains employed in the project till a later date. It can be used as a good indicator of the resource release date in the project. Simultaneous employment of M x and M y in optimization procedure brings about better utilization of available resources and increases the applicability of the model in management of real world problems. The GAs technique was used to detere the optimal solution. In addition, for handling multiobjective optimization, weighted sum method was utilized. The results of the model demonstrated that required resources for critical activities were very important in deteration of the modified total duration of project. When the amount of resource utilization for options of critical activities was high, the model tended to select the option with lower required resources to decrease the project total duration while it may cause project total cost increases.. References Harris, F., and McCaffer, R. (). Worked examples in construction management. Granada. Hegazy, T. (). Optimization of resource allocation and leveling using genetic algorithms. Journal of Construction Engineering and Management, Vol., No., pp -. Hegazy, T., and Ersahin, T. (). Simplified spreadsheet solutions. II: overall schedule optimization. Journal of Construction Engineering and Management, Vol., No., pp -. Kandil, A., and El-Rayes, K. (). MACROS: multiobjective automated construction resource optimization system. Journal of Construction Engineering and Management, Vol., No., pp -. Karshenas, S., and Haber, D. (). Economic optimization of construction project scheduling. Journal of Construction Management and Economics, Vol., No., pp -. Li, S. (). New approach for optimization of overall construction schedule. Journal of Construction Engineering and Management, Vol., No., pp -.