# Multi-Objective Resource-Constrained Scheduling in Large and Repetitive Construction Projects
## 4 Case Study

Table 3: Optimal results for example project. but not the cumulative excess along the project. In addition, because the number of resources appears in integer form, several discrete solutions present the same MaxR value (but different RLEs), making it difficult to differentiate among them in terms of detailed effectiveness;
* There is no single criterion among these three that clearly outperforms the other two in all or most cases. The results in Table 3 show that a parameter can obtain a better value when another objective is pursued. This means that it is rather useful to experiment with all three criteria within the analysis process;
* The RIO criterion is not closely aligned with the goals of resource diagram smoothness and resource overallocation minimization. It rather pursues to develop resource allocation patterns with few and low transitions in resource usage from day to day. As a result, it generally permits higher resource overallocation but smoother transitions from level to level in time. The solutions are usually associated with reduced project duration and higher average resource loading. In this case, some extra cost savings can be realized in terms of indirect project costs from duration reduction (not considered in the analysis). Similarly to the RLE criterion, the RIO can be practically valuated in pure cost terms if the unit cost of the resource (primarily machinery type) and movement in and out of the project is known;
* The employment of multi-objective or sequential optimization appears to provide solutions that are often of higher quality than single-objective optimization. In every optimization case, all four parameters associated with the final solution are recorded so that the real-life cost of this solution can be calculated;
* The employment of multiple runs with the same criterion (or criteria) appears also to be beneficial as the stochastic nature of the solution method leads to distinct solutions among alternative runs;
* A useful tip for performing optimization when the objective function takes discrete values (this pertains to RLE, RIO, and mostly MaxR, which obtain integer values within a short range) is the following. The evolutionary process generates and evaluates new solutions as the process goes on. If these new solutions fail to push the fitness value to its next discrete level, the solution population is not promptly and adequately refreshed to effectively search the decision space. To avoid this trapping, one can add a small perturbation in the objective function to let the optimization engine be continuously active. In the present case, a very small STD component (appropriately regulating the weight w4) and not practically affecting the main optimization criterion is added to the objective function to provide such kind of service. In fact, the majority of the results presented in Table 3 (best values in each case) have been obtained with such a formulation. The above augmentation is not useful in cases where the optimization parameter already calculates continuous values (e.g., STD). To obtain a more global picture of the algorithm's effectiveness in achieving the desired outcome, the best parameter values from Table 3 are presented in relative terms in Figure 13. In particular, the RLE and RIO values are related to the total number of resources used throughout the project while the MaxR and STD values reflect deviations fromthe average resource level in each case. The results indicate that the total number of resources in excess of resource availability (RLE) and the total number of the resource movements in and out of the project (RIO) are in the order of 3% and 7.5% in relation to the cumulative number of resources used throughout the project. The standard deviation of the resource usage is between 10% and 20% in relation to the corresponding mean value. Finally, the maximum resource usage overruns the mean value by 25% to 33%. All deviations tend to increase as the project length increases and the mean resource usage decreases.