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In real-world environments, selecting the right contractor is an important issue which considerably influences on completion time (makespan), total cost and quality of performing the project. This paper presents a framework to deal with the multi-mode resource constrained project scheduling problem (MRCPSP) and contractor selection (CS) in an integrated manner. The presented problem is called MRCPSP-CS, in which each activity is assigned to a contractor, an execution mode is selected for each activity, and the start/finish times of project activities are determined. The present paper presents a discrete-time (DT) bi-objective optimization model to deal with MRCPSP-CS, aiming to minimize the total cost and completion time of the project, simultaneously. Then, four multi-objective decision making (MODM) techniques, namely individual optimization, max-min, LP metrics and multi-choice goal programming with utility functions are used to solve the proposed model. Computational results show that the multi-choice goal programming with utility functions, LP metrics and max-min methods outperform other MODM techniques in terms of average makespan, average total cost and average CPU time, respectively. Hence, Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) is used to assess the performance of MODM techniques. Results of applying TOPSIS show that in small-, medium- and large-sized test problems, multi-choice goal programming with utility functions outrank other MODM techniques.
Keywords: Multi-mode Resource Constrained Project Scheduling Problem (MRCPSP), Contractor Selection (CS), Multi-objective decision making (MODM), Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS)
Over the past few decades, research on the classical resource-constrained project scheduling problem (RCPSP) has been extensively expanded. In the classical RCPSP, a number of activities are scheduled, such that some objectives (in the form of time, cost, quality, etc.) are optimized, considering the limited resource availabilities and precedence relations (Hartmann and Briskorn 2010; Maghsoudlou et al. 2016; Xiao et al. 2016).
Generally, RCPSP models are classified into discrete time (DT) and continuous time (CT) models (Kopanos et al. 2014). In DT models, events can be started/finished in a finite predefined set of time points. On the contrary, in CT models, start/finish times of events can take place at any time point in the time horizon. (Kone et al. 2011).
Kaplan (1988) was the pioneering work that presented a discrete time binary integer programming (DT-BIP) model to deal with RCPSP in both single-project and multi-project settings. Kaplan (1988) formulated a DT-BIP formulation for the preemptive version of the RCPSP, which can be easily adopted to the classical RCPSP, as shown by (Klein 2000). Kaplan (1988) presented a single kind of binary variables that determine whether activity i is active in time period t or not, allowing to define a simpler definition of resource constraints. Klein (2000) presented two DT-BIP models to deal with RCPSP. The first formulation is based on the definition of binary variables that determine whether activity i starts at the beginning of time course t or earlier. The second formulation is an improved version of the prior formulation with (I) a different range for the above-mentioned binary variables, and (II) novel binary variables that explain whether an activity i is completed at the time course t or earlier. Alvarez-Valdés et al. (1993) proposed a continuous time mixed integer programming (CT-MIP) formulation based on minimal resource incompatible sets. These sets consist of activities that have no prerequisite relations, but cannot be performed together due to the limited resource availability. The optimization model is formulated based on (I) binary variables that determine the sequence of activities, and (II) integer variables that demonstrate the start time of activities. During the last three decades, several studies have been carried out using DT-BIP models, e.g. Mingozzi et al. (1997), Bianco et al. (2013), Kopanos et al. (2014), Rostami and Bagherpour (2017), Chakrabortty et al. (2018) and Shahsavar et al. (2018), and CT-BIP models, e.g. Koné et al. (2011), Kyriakidis et al. (2012), Kopanos et al. (2014), Alipouri et al. (2017) and Naber (2017), to deal with RCPSP and its extensions.
The Multi-Mode Resource-Constrained Project Scheduling Problem (MRCPSP) is an extension of the basic RCPSP, in which each activity may be performed in more than one execution mode. An execution mode makes a tradeoff between resource usage and processing time needed to perform an activity. In fact, in MRCPSP, mode selection decisions are added to those made in RCPSP (Afshar-Nadjafi 2014; Beşikci et al. 2014). (Elloumi and Fortemps, 2010) formulated a bi-objective optimization model to deal with the multi-mode resource-constrained project scheduling problem (MRCPSP). In addition, they presented an evolutionary algorithm that allows the violation of non-renewable resources using a penalty function. (Nabipoor Afruzi et al., 2014) presented a multi-objective optimization model to cope with a multi-mode Discrete Time-Cost-Quality Tradeoff Problem (DTCQTP). Gutjahr (2015) presented a bi-objective optimization model to deal with MRCPSP, aiming to minimize total costs and makespan of projects. (Maghsoudlou et al., 2016) proposed a multi-objective Time-Cost-Quality optimization model for the multi-skill multi-mode resource constrained project scheduling problem.
(Elloumi, et al., 2017) proposed a number of new disruption measures to optimally schedule activities in a multi-mode resource-constrained project scheduling problem, considering the project makespan and a disruption measure as objective functions. They used a penalty to handle the potential violation of nonrenewable resources.
A key step of managing a project is to select the right contractor who can perform the project in a timely, low cost, and high quality manner. This step plays an important role in the success of the project (Namazian and Yakhchali 2016). Generally, in today’s complex environments, contractor selection is a challenging issue, since it needs to make many trade-offs involved in the decision making process. The wrong choice of the contractor may increase the chance of rework, project costs and completion time (Turner 2015; Zavadskas et al. 2017). Due to the important role of selecting the right contractor on the cost, quality and completion time of a project, many studies deal with the contractor selection problem. However, to the best of our knowledge, none of studies consider the interrelated nature of project scheduling and contractor selection problems. In fact, decisions on schedule of activities and assignment of activities to contractors are made in a disjointed, consecutive manner. This may lead to decisions which are suboptimal to a holistic view that makes all interrelated decisions in an integrated manner. Hence, different from other studies, this study considers an integration of Multi-Mode Resource-Constrained Project Scheduling Problem (MRCPSP) and Contractor Selection (CS) problem. More precisely, a bi-objective optimization model is proposed that makes decisions on start times of activities, execution modes of activities and assignment of activities to contractors in an integrated manner. Afterward, a number of Multi-Objective Decision Making (MODM) techniques are adopted to solve the proposed bi-objective optimization model, and results are compared and analyzed.
The structure of the research is as follows. In section 2, a description and mathematical formulation of the problem is proposed. In section 3, a number of MODM techniques are proposed, as solution methodologies, to solve the proposed bi-objective optimization model. An extensive numerical analysis is presented in section 4. Finally, section 5 concludes the paper.
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