# Multi-Objective Resource-Constrained Scheduling in Large and Repetitive Construction Projects
## 3 Proposed Model
These include the following:  
* RIO--Resources in and out: this accounts for the sum of resource unit deviations from day to day and represents the cost of moving resources (human and machinery) in and out of the construction site day by day. \[RIO=\sum_{t=1}^{D+1}\lvert R_{t}-R_{t-1}\rvert,\,\,R_{0}\,\,=\,\,R_{D+1}\,\,= \,\,0,\] (5)
* RLE--Resource limit excess: this measures the cumulative number of resources exceeding the daily resource availability and accounts for the financial impact of recruiting more additional resources than initially planned. \[RLE=\sum_{t=1}^{D}(R_{t}-R_{m}),\,\,\forall\,\,t\,\,\text{at which}\,\,R_{t}\,\,>\,\,R_{m},\] (6)
* MaxR--Maximum resource demand: this shows the maximum resource value in the resource histogram indicating the cost from attempting to satisfy the full resource requirements. \[MaxR\,\,=\,\,max_{t=1}^{D}R_{t},\] (7)
* STD--Standard deviation: the standard deviation of the resource histogram values in time is considered as well for the sake of comparison. \[STD\,\,=\,\,\sqrt{\frac{\sum_{t=1}^{D}(R_{t}-R_{m})^{2}}{F_{N}-1}},\] (8)  
The above parameters (decision criteria) are incorporated in a weighted-sum objective function of the following form:  
Figure 1:0  
where \(w_{i}\) accounts for the corresponding unit cost values, defined by the user and representing the problem characteristics. With such a formulation, the distinct criteria can be evaluated both individually and collectively. The weights in the above formulation indicate the importance of each parameter in any specific case. Regarding the physical meaning of the weights, RLE and RIO account for parameters that can be directly valuated in cost terms of the corresponding resource allocation inefficiencies, thus, the related weights hold unit cost values. MaxR and STD are generic parameters that are often used to smooth out temporal deviations of such histograms. For these parameters, the weights are appropriately normalized to provide the desired level of influence in a bi- or multi-objective assessment. In real-life (construction) projects, scheduling and resource utilization decisions are most probably assisted by simple formulations and handy tools that can be easily used by design engineers. With this aim, the RCSP problem is developed in an Ms-Excel 365 spreadsheet structure to assist development, readability, model performance checking, and result evaluation. The employment of existing commercial software for optimization significantly alleviates the reluctance of the engineer to dig in and develop an optimization algorithm or implement one from the literature. While general-purpose software may not be as effective as a fine-tuned algorithm for a specific problem, it is still a valuable tool to significantly improve resource allocation, especially in large projects. Genetic Algorithms have been found to be effective in resource-constrained scheduling problems, presenting a stable growth since 2010 and being the most used in the last five years [37]. Further, commercial evolutionary algorithm-related optimization software has principally incorporated this type of algorithm. Optimization in this study is performed via a commercial optimization software (Palisade (now named Lumivero) Evolver, v8.6.0, [[https://lumivero.com/products/decision-tools/evolver/](https://lumivero.com/products/decision-tools/evolver/)]([https://lumivero.com/products/decision-tools/evolver/](https://lumivero.com/products/decision-tools/evolver/)), accessed on 1 August 2024), which runs as an add-in of Ms-Excel. The software allows different parametrization of the GA structure. As part of this analysis, it was found that, although this parametrization has not displayed much impact on the solution quality, a formulation with 50-chromosome population size and crossover and mutation rates of 0.5 and 0.1, respectively, seems to provide somewhat improved results. The problem runs have been performed on a typical PC with the following specifications: Intel(r) Core (TM) i3-3110M CPU @ 2.4 GHz, RAM 4 GB.