An Applicable Method For Resolving Resource Over-Allocation Problem While Using PERT
In MRCPSP problems activities can be scheduled in more than one way and therefore the activities might have different durations and resources and consequently cash flows. (Węglarz et al. , 2011) published a review paper in regard with the literature of the MRCPSPs. (Laslo, 2010) proposed an new method for minimizing negative dependent cash flows in scheduling process. Financial aspect of scheduling projects should not be ignored as they may cause providing infeasible or useless schedules. It should be mentioned that in MRCPSP studies there are 2 main cash flows can be considered. Positive cash flows that are referred to earned money of the project. Negative cash flows in contrast are those costs that should be considered for completing projects such as expert salary, worker’s wage, and maintenance and services costs. Cash flows whether they are positive or negative can affect activity due date, completion time, resource availability, material purchasing etc. (Yu et al. , 2012) employed genetic algorithm for selecting multi-criteria project portfolio problem. Their goals were project interactions and preference information.
Afterward, Kolisch & Drexl (1997) also proved found that increasing number of resources causes increasing in the degree of complexity of MRCPSPs. Hence, most of the researches in this area used heuristic methods for solving the problems. (Yan et al. , 2009) focused on finding a solution to prepare a fast response to maritime disasters using heuristics. Metaheuristics are also used in many cases for solving MRCPSP problems. Among all metaheuristics genetic algorithm is used more frequently than others Alcaraz et al. (2003; Hartmann (2001; Lin & Gen (2008; A Lova et al. (2006; Mori & Tseng (1997; Ozdamar (1999). (Kim et al. , 2005) used fuzzy based genetic algorithm for minimizing completion time and tardiness penalty. Naber & Kolisch (2014) proposed 4 RCPSP based profiles discrete-time mixed integer programming models with flexible resource. To solve the model they offered preprocessing and priority-based heuristic methods. (Ke and Liu, 2010) also used a combination of fuzzy and genetic algorithm for optimizing project cost respecting to completion time (see also Chen & Askin (2009); Hartmann & Briskorn (2010).
Particle swarm optimization method is also used by (Jarboui et al. , 2008) for solving MRCPSPs. 2. 3 Maximizing Profit of the project (Profit/Net Present Value (NPV))In many researches in contrast, the aim were to increase the benefit of scheduling activities. Each of the activities has positive and negative cash flows that can be changed by executing mode of activity, scheduling time and different types of resource. Maximizing NPV is logic way to choose a project or reject it. If NPV of a project is calculated and the result is a negative value then it can be concluded that the project does not provides any benefit. (Russell, 1970) developed a resource constraint model where the objective was maximizing the NPV. Afterward, (Grinold, 1972) converted the model proposed by Russell into linear model and developed two optimal finder algorithms considering fixed and variable due date of project. (Elmaghraby and Herroelen, 1990) developed a resource constraint algorithm for maximizing NPV. Sung & Lim (1994) proposed a heuristic to improve the NPV by decreasing the durations of activities.
Aidin Delgoshaei & Gomes (2016) proposed a new method for scheduling in the presence of uncertain cost. Laslo (2010) considered negative cash flows to minimize the project completion cost as objective function of their model. (Etgar et al. , 1997) showed that resources beyond time limit can have significant effect on makespan of project Meanwhile, (De Reyck, 1998) proposed a new method for scheduling activities by considering both positive and negative cash flows. Their objective was minimizing completion time. (Icmeli et al. , 1993) discussed that adding resources limitations caused turning model into a non-poly nominal model which cannot be solved easily by optimizing algorithms. Then, they considered discounted rate in the proposed a model a way that more cash flows will be earned in case of completing an activity in shorter period (RCPSPDCF).
Afterward, many researchers tried their utmost effort with the aim of solving the problem of maximizing NPV while discount rate is taken into consideration. (Baroum and Patterson, 1999) developed a model for maximizing discounted NPV with 50 variables. Afterward, (Icmeli and Erenguc, 1994) used Tabu search (TS) algorithm in solving RCPSPDCF problem. They set penalty for activities later than the due date. (Yang et al. , 1993) developed statistical programming for scheduling projects with positive cash flows problems. In continue, Zhu & Padman (1999) Tabu search is also used as an effective way for scheduling discounted cash flows resource constraint problems. Mika et al. (2005) developed a mathematical model renewable and non-renewable resource constraint scheduling method. Their objective was maximizing NPV which was solved by a hybrid simulated annealing and Tabu search algorithms. (Aidin Delgoshaei et al. , 2014) proposed a new mathematical model for maximizing the net present values of the project while discounted positive cash flows are taken into account. Their model only considered finished to start relations.
Recent researches (after 2005) many researchers considered preemptive resource for showing the priority of activities in using resources. Aidin Delgoshaei et al. (2016a) focused on scheduling dynamic manufacturing systems using a hybrid genetic and simulated annealing algorithms. (Demeulemeester and Herroelen, 1996) developed a method for scheduling preemptive resource constraint problems. (Buddhakulsomsiri and Kim, 2006) argued that preemptive resources can affect to makespan of the project and hence it must be considered during scheduling problems. (Damay et al. , 2007) used LP method where the RCPSP has preemptive resources. In continue (Ballestín et al. , 2008) developed a heuristic algorithm in similar problem. Seifi & Tavakkoli-Moghaddam (2008) solved the maximizing NPV problem in 4 different payment methods. (Van Peteghem and Vanhoucke, 2010) developed a new method for minimizing completion time in an activity split allowed multi-mode resource constraint method. Aidin Delgoshaei et al. (2016d) proposed a new method for maximizing NPV in the multi-mode resource constrained scheduling problem. The aim of their model is to prevent resource over-allocation during scheduling process.
In their model all relation types between activities are considered. Reviewing the literature of project scheduling problems show that maximizing NPV of the MRCPSP’s while durations of activities are uncertain is not developed. Moreover the negative cash flows are mainly not considered in maximizing NPV. Hence this research continues Aidin Delgoshaei et al. (2016d) research as base-paper and continues their research by considering uncertain durations of activities, considering contracted time as a constraint for providing schedules, considering contracted cost as a constraint for providing schedules. Table 1 shows the characteristics that will be developed in this research in continue of opted papers in Section 2: In the next Sections (Sections 3 and 4) a mathematical programming method will be developed first. Then, an appropriate solving method will be proposed to solve the model. Results are then evaluated with some performance measurements.
Research Methodology
Section 3 explains the methodology of the research in details. The outcome of the literature review indicates that most of the researches in Section 2 used mathematical modelling for expressing the problem statement. Hence, a non-linear mixed integer programming method is developed. Then, the model is analysed and some information about data gathering, solving machine (computer) and solving algorithms are explained.
Develop an Appropriate Model
In this part, a new mathematical model will be developed where the aim is to find the impact of uncertain activity duration is maximizing the NPV of the MRCPSP’s while both positive and negative ash flows are taken into consideration. This research is done in continue of Aidin Delgoshaei et al. (2016d) by developing some new ideas. In base-paper (Aidin Delgoshaei et al. , 2016d) the durations of activities are considered fixed but we decided to continue that research with stochastic durations that mean the model is not a simple critical path method anymore and instead after this change the model is according to stochastic method. The objective function of this model is contains calculating NPV using positive cash flows and negative cash flows. The term e^((α/t)) is a famous formula in engineering economy while the interest rate is calculated continuously. Hence the term ∑_(t=1)^TH▒∑_(i=1)^n▒∑_(m=1)^M▒X. 〖PCF〗_((i,m,t)). e^((α/t)) is used to calculate the NPV for all activities in every scheduling period. Similarly, negative cash flows are calculated in the same logic. The model is developed in a way that activities can be scheduled in remained resources areas in every single day.
Hence, the algorithm divided activities into smaller pieces considering their duration and then allocates the single the activities through the calendar of the project. The first constraint is to show the beginning of project by setting the day 1 foe early start of the first activity. The second constraint shows the early start of the rest of the activities. The third constraint is used to show the finish to start relation between activities. Similarly, equations 13, 14 and 15 are used to explain the SS, FF and SF relations respectively. The equation number 16 is set to ensure that none of schedules are exceeded than the completion time that is signed in the contract. Next constraint indicates that number of the scheduled days for an activity must not be exceeded than its estimated duration.
The equation number 18 is set to show that an activity can be choosing only one mode during its execution. The 9th constraint is used to prevent the algorithm from scheduling activities in over allocated days. The 10th constraint is developed to ensure that all feasible solutions have total cost less than contract cost. The equation number 21 is developed to show that durations of activities must be estimated using triangular probability function. The last constraints show that the X must be binary and ES variables must be integer.
