# Multi-Objective Resource-Constrained Scheduling in Large and Repetitive Construction Projects
## 4 Case Study
Figure 10: Resource histogram for STD-RIO criterion. The red line indicates the daily resource availability. Figure 9: Resource histogram for RIO criterion. The red line indicates the daily resource availability. Table 3 presents the input and output values of the analyzed cases. Case 0 refers to the initial (non-optimized) project schedule corresponding to the early start times. This schedule acts as the initial optimization solution in all cases examined. Case 1 represents the solution with the minimum standard deviation (STD) of the daily resource usage as a single objective. Similarly, Case 2 provides the result of minimizing the cost of exceeding the daily resource availability (RLE). In the same path, Case 3 indicates the solution of minimizing the maximum resource demand (MaxR) that is observed during the project. The last single-objective Case 4 examines the minimization of resource movements in and out of the project throughout its length (RIO). The next two cases present the results of bi-objective optimization; specifically, Case 5 considers the criteria STD and RIO, and Case 6 the criteria RLE and RIO. Finally, Cases 7 and 8 incorporate two-step sequential optimizations between criteria STD and RIO and RLE and RIO, respectively, meaning that an optimization run is performed with respect to the first criterion and, upon the optimum solution found, another optimization is run with respect to the second criterion. The rationale behind such development is as follows. The problem under consideration aims to provide real-life solutions in regard to actual construction (or similar) projects. In this direction, there are two main cost contributions, one from exceeding the resource availability and the other related to the need for transferring resources in and out of the project. In extreme cases, only one of these cost elements may exist. For instance, there may be the case that it is costly to recruit additional resources (e.g., use of special machinery) but the cost of moving such resources is negligible (e.g., sub-projects being next to each other). In such a case, a rather single-objective project is encountered, using either of the three parameters, STD, RLE, or MaxR, which serve this objective (Cases 1, 2, and 3). On the other hand, there may be cases in which additional resources can be easily allocated to the project with no extra cost for overallocation (this mostly concerns human resources of an ordinary specialty). In addition, the cost in and out of the project may be high, for instance, if the sub-projects are dispersedly located. In the general case, both objectives provide some contribution to the resource allocation cost so that optimization of both objectives, in the form of a cost-sum approach, is sought based on their relative unit cost to each other (Cases 5 and 6). The last two cases (7 and 8) aim to investigate whether the optimization of the two conflicting objectives in a serial mode of respective single-objective optimizations can provide better results compared to the concurrent optimization of the two parameters. The analysis has been extended to several levels of project durations and corresponding resource thresholds. In each case, the RCSP has been considered, employing individual criterion or pairs of criteria and optimization modes (single- or two-step optimization). In each run, all parameter values that correspond to the final solution are recorded. Each optimization case is run three times to account for the expected variability in results due to the stochastic nature of the optimization (GA) algorithm. Among the three outputs, the best one is registered. This is an anticipated course of action in practical applications of such a problem. In a more research-oriented effort, the typical course of action is to perform several trials and record the statistics of all runs (average or median, standard deviation,  
Figure 12: Resource histogram following a 2-step optimization process (STD-RIO). The red line indicates the daily resource availability. confidence intervals, etc.).