Optimal crashing time scheduling for mega projects

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This Paper presents a framework for time crashing a mega project by using the linear programming technique such to earned least total crashing cost.

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International Journal of Management (IJM) Volume 9, Issue 2, March–April 2018, pp. 112–122, Article ID: IJM_09_02_013 Available online at http://www.iaeme.com/ijm/issues.asp?JType=IJM&VType=9&IType=2 Journal Impact Factor (2016): 8.1920 (Calculated by GISI) www.jifactor.com ISSN Print: 0976-6502 and ISSN Online: 0976-6510 © IAEME Publication OPTIMAL CRASHING TIME SCHEDULING FOR MEGA PROJECTS Rami A. Maher Isra University Amman, Jordan Ibrahim Abed Mohammad Isra University Amman, Jordan Hamzeh H. Al-Safadi Isra University Amman, Jordan ABSTRACT This Paper presents a framework for time crashing a mega project by using the linear programming technique such to earned least total crashing cost. First, a mathematical model of the problem is given, and then after a numerical example is introduced to demonstrate the efficiency of this technique in solving the time crashing problem of mega projects for different time crashing values. This will help the project managers to consider strategic decisions for crashing the milestones and activities constituting the mega project. Key words: Crashing Scheduling, Linear Programming, Mega Projects, Milestones Cite this Article: Rami A. Maher, Ibrahim Abed Mohammad and Hamzeh H. Al- Safadi, Optimal Crashing Time Scheduling for Mega Projects, Practices and Management, International Journal of Management, 9 (2), 2018, pp. 112–122. http://www.iaeme.com/IJM/issues.asp?JType=IJM&VType=9&IType=2 1. INTRODUCTION Mega project composed from two words “Mega” and “Project”, the word Project which identified as temporary endeavor undertaken to create a unique product, result or service [1]. Meanwhile, the word “Mega” is to determine that something is very huge. Thus, mega projects are temporary endeavor undertaken characterized by large investment, vast complexity (especially in organizational terms), and long lasting impact on the economy, environment and society [2]. Mega project includes airports, power plants, tunnels, large-scale sporting events and natural gas extraction that are examples from very wide number of mega projects. As a general rule of thumb “ Mega projects” are measured in billions of dollars , “ Major projects” in hundreds of millions and “Project” in millions and tens of millions [3]. One of the http://www.iaeme.com/IJM/index.asp 112 editor@iaeme.com
Optimal Crashing Time Scheduling for Mega Projects most important properties is that the mega project cannot be delivered on time because of the complexity and huge number of activities that should be considered by the manager. The question arias now is what is Mega project ? To answer this question many dimension must be discussed. For instance, from investment point of view, projects have capital cost more than 1$ billion dollars are considered as mega projects [4]. The problem with mega projects is the frequent failure of many of these projects. For example, Analyzing a dataset of 318 industrial mega project shows that 65% of them are classified as failure [5]. Another example, Oil and gas production sector is the worst, where 78% of the mega projects categorized in failure area. A major cause of this failure is that the actual costs of these project are higher than estimated costs and actual implementation schedule was higher than estimated schedule[6]. Van Merrewijk et al. define mega project as “multibillion-dollar mega infrastructure usually commissioned by government and delivered by privet enterprise; and characterized as uncertain, complex, politically- sensitive and involving large number of parties [7]. In what follows in this introduction, some fundamental concepts will be introduced. In general Project management is the discipline of initiating, planning, executing, controlling and closing the work of team to achieve specific goals and meet specific success criteria. Project in general consist from activities which identified as smallest identifiable and measurable piece of works can planned and managed. Activities start with events and end with events. There are several methods to represent projects activities used such as the activity on arrow AOA and activity on node AON. For project scheduling two analytical approach are usually used; the critical path method CPM and project evaluation and review technique PERT. The major different between these two approach is that the PERT technique utilizes uncertainty to develop activities duration. One important process in project management issue is the analysis of possible time- crashing for an assumed additional resources. Project crashing and cost law optimization generally was studied in several times to add new information to existing body of knowledge, these study developed for determining the least scheduling deadline for the project with least costly crashing schedule to make optimum solution [8]. The simulation approach for optimization project cost and schedule is one of tools that can be used to bring back the project under control. The simulation approach consists a collaboration between project manager and IT department in company to make optimum scheduling time and cost overruns. The optimization of time and cost process could be considered as standard process for each project separately; the time spent on the actual crashing was minimal and the project scheduling time must be reduced to the minimum level of the optimization of saving money and time [9]. Linear Programming (also called linear optimization) is a special technique from mathematical optimization that aims to achieve best outcomes in minimal or maximal sense from a linear relationship of many variables that are simultaneously subjected to linear equalities and/ or inequalities constraints. The linear programming technique has been used efficiently to solve several problems in project management including the crashing problem, [10]. This paper is devoted to give and illustrate a methodology of optimal time crashing of a mega project by utilizing the analysis of milestones characteristics and linear programming technique. http://www.iaeme.com/IJM/index.asp 113 editor@iaeme.com
Rami A. Maher, Ibrahim Abed Mohammad and Hamzeh H. Al-Safadi 2. PROBLEM STATEMENT Let a certain mega project consists of milestones whose time interrelationship network is given, and that the normal completion time of the mega project is already calculated together with one set of critical activities, i.e. a single critical path. Furthermore, each milestone has its predecessor list of activities, normal duration , normal cost , and the crashed duration and corresponding crashed cost of each activity. It may be the case that some activities within the milestones cannot be crashed; i.e. they have a crashed duration equal to the normal duration. Consequently, the crashing slop for each critical activity within each milestone is calculated from the following relation The task is to crash the mega project to a certain completion time less than by a specified amount of units of time, say with minimum additional cost (additional resources). The amount of crashing can be obtained either from a specific subset of the critical milestones or from the addition of possible individual crashing time amounts from each critical milestones. obviously, this will depends on the original scheduling of the mage project itself. 3. METHODOLOGY AND MODELING The scheduling analysis of a mega project requires a general software tool to deal with different tasks within the solution framework of the scheduling process. It is known that most scheduling operations desire the activity network of each milestone, i.e. to have the AOA network. This facilitates the use of the mathematical optimization efficiently. Therefore, the first step is to obtain the time matrix and the number identifier matrix of activities of each milestone. These two matrices are given by the following definitions { { An algorithm that performs the generation of these two matrices directly from the given predecessor list of activities can found in [11]. The next step is to determine the critical path value and the activities along this path . One efficient method to do that is to utilize Floyd's algorithm [12] with some simple modifications. This algorithm provides also the earliest starting time and the latest starting time and consequently the free and the total floats for each activity [13]. Next, the crashing slops are computed and stored in corresponding to each activity. These three steps are performed for all milestones of the considered mega project. Finally, the possible largest crashing time of each milestone will be obtained, where ∑ where is the number of the critical activities in the milestone (excluding any critical activity that it cannot be crashed for any reason). Noting that in case of multi critical path problem, for primary investigation, the crashing calculation should consider only the critical activities that are common for all critical paths. Then the maximum possible crashing time of the mega project is given by http://www.iaeme.com/IJM/index.asp 114 editor@iaeme.com
Optimal Crashing Time Scheduling for Mega Projects ∑ Consequently, the required corresponding maximum additional cost that perform the can be calculated. Now if , then the manager has no choices but only to consider the above calculations, and crashing individually each milestone . However, if then the manager will have multi choices for achieving the desired crashed completion time of the considered mega project. For the latter case, the linear programming model is proved to be a suitable technique for obtaining optimal results of such a task. The linear programming model for the crashing problem can be stated as follows: 1. Define a set of variables to denote of the amount of units of time that each activity will be crashed, where . In addition, another set of variables has to defined; it contains the starting times of activities, . The starting time of this activity is computed from 2. To define the cost function to be minimized, i.e. ∑ where is the crashing slop associated with activity. 3. To define a set of constraints including crashing time constraints, nonnegative constraints, and unfolding the network constraints (every activity should be start after preceding activity completely finished); these are defined as ( ) ( ) where are linear relations of starting time of network activities, is a number depends on the predecessor list of the project, and are time values conducted from the normal duration of the network. The methodology of applying the linear programming technique for an optimal crashing time of a mega project will be performed in two phases. In the first stage, the mega project network will be analyzed to obtain the optimal amount of crashing time that should taken from the completion time of each critical milestone, then, in the second phase, these amounts are optimally distributed for crashing the critical activities of each milestone. 4. SIMULATION AND RESULTS For illustrating the methodology of the time crashing problem of mega projects, a simple five milestones mega project will be considered. The network of the project is very simple, it is just a sequential series of the five milestones; this simply means that the all milestones are on the critical path and the completion time is simply the addition of durations. http://www.iaeme.com/IJM/index.asp 115 editor@iaeme.com
Rami A. Maher, Ibrahim Abed Mohammad and Hamzeh H. Al-Safadi Table 1 summarizes the initial data of these milestones including the normal completion time (duration) and the total required normal cost of each milestone. Table 1 Initial data of milestones Normal Milestone Number of Normal Cost Duration Name activity $ (days) 22 248 1406900 20 220 1208600 48 300 3326250 50 340 4035176 60 400 4100735 Then the total normal completion time of the mega project is equal to 1508 days and a total normal cost equal to 13,977,661 $. As a sample of computation, milestone will be considered; it has an activity on node network AON shown in figure 1, and a predecessor list given in table 2. As it can be noted that activities cannot be crashed for some reasons. In table 2, it is also given the possible crashing duration of each activity and the crashing cost to perform the crashing. Figure 1 Activity network of milestone The information in table 2 is used to calculate the crashing slop associated with each activity. However, it can be shown easily that milestone has six critical paths, and for a certain time crashing, such characteristic should be considered carefully. Table 3 gives the crashing time in days, the crashing slops, one of the possible critical path. However, the maximum possible crashing time of this milestone is equal to 20 days; common critical activities are only considered. In similar way, the same information can be obtained for all other milestones of the considered mega project. Table 4 summarizes these calculation for each milestones. From data of table 4, it can be concluded that is 20 days and that sums 128 days as a possible maximum value of crashing time. http://www.iaeme.com/IJM/index.asp 116 editor@iaeme.com
Optimal Crashing Time Scheduling for Mega Projects Table 2 Detail information of milestone Normal Crashed Activity Prede duration duration Normal Crashed Name cessor Cost $ Cost $ (days) (days) - 15 13 45000 49000 15 14 43500 44850 15 13 46300 48840 15 12 54000 56830 15 13 53700 57530 18 17 43200 44900 18 16 75600 80800 18 18 75600 75600 20 18 108000 115000 20 18 72000 89570 20 18 144000 151200 20 18 36000 38520 20 19 48000 50688 20 19 72000 74160 20 19 136000 144160 20 20 108000 108000 10 10 18000 18000 10 9 42000 45780 18 16 75600 79834 10 9 24000 24480 12 11 86400 89150 0 0 0 0 Table 3 Crashing of milestone A Crashing Crashing Activity slope Time Calculation name (Days) $/days (days) 2 2000 1 1350 2 1270 For common critical activities 3 943 2 1915 1 1700 2 2600 0 0 2 3500 2 8785 http://www.iaeme.com/IJM/index.asp 117 editor@iaeme.com
Rami A. Maher, Ibrahim Abed Mohammad and Hamzeh H. Al-Safadi 2 3600 2 1260 1 2688 1 2160 1 8160 0 0 0 0 1 3780 2 2117 1 480 1 2750 0 0 Table 4 Crashing summery of all milestones Total Total Milestone Normal Crashed Normal Crashed Name Duration Duration Cost Cost 248 228 1406900 1486900 220 188 1208600 1243263 300 286 3326250 3402200 340 310 4035176 4126676 400 369 4100735 4223757 Therefore, the manager can crashed the mega project maximally to 1382 days (i.e. a crashing time equal to 126 days) and he has to pay an additional cost of 505135 $. However, if the manager wants to crash the mega project to say 1400 days (i.e. a crashing time equal to 108 days), then he has to determine optimally the crashing schedule among the milestones and within each milestone. For this purpose, the linear programming technique will be used first to determine optimally the contribution of each milestone to sum the required crashing time. Considering the data of table 4 and by calculating the crashing slop of each milestone, the linear programming model becomes Subjected to the constraints http://www.iaeme.com/IJM/index.asp 118 editor@iaeme.com
Optimal Crashing Time Scheduling for Mega Projects Note that in this simple series connected milestones, the number of time constraints is equal to the number of milestones, and the values of are the normal durations of the milestones; this is may not be the case with different milestone network. The model can be solved by utilizing the LINDO 6.1 optimization software. The optimal integer solution is feasible and the results are as shown in table 5. The results tell that milestone is crashed to a value less than the value by 5 days and milestone is not crashed at all, while milestones are crashed up to their maximum possible values. The optimal crashed cost of mega project is calculated as Thus, the optimal (minimum) additional cost is 57460 $ due to crashing time of 108 days. Table 5 Linear programming results of mega project Variable Crashing time (days) 15 32 0 30 31 0 233 421 721 1031 1400 Next, the manager is left only with applying again the linear programming model to milestone to determine the crashed activities that sums the 15 days. Considering the data of tables 2 and 3, the following linear programming model is proposed Subjected to the constraints http://www.iaeme.com/IJM/index.asp 119 editor@iaeme.com
Rami A. Maher, Ibrahim Abed Mohammad and Hamzeh H. Al-Safadi The integer solution of the above linear programming model is feasible, and the optimized results are given in table 6. As shown in the table 6, only 10 activities of the milestone are crashed to sum the 15 days. All these crashed activities are crashed by their maximum possible crashing time except activity , which is crashed by half of its maximum possible crashing time. However, for this critical multipath milestone, the activity has to be crashed by one day to keep the critical path of milestone equal to 233 days as desired. Therefore the total additional minimum amount of cost to crash milestone is 37327 $. Table 6 Optimal crashing results of milestone Crashing time Additional Activity name of milestone (days) crashing cost $ 0 0 1 1350 0 0 3 2830 2 3830 0 0 0 0 0 0 2 7000 1 8785 http://www.iaeme.com/IJM/index.asp 120 editor@iaeme.com
Optimal Crashing Time Scheduling for Mega Projects 0 0 0 0 1 2688 1 2160 0 0 0 0 0 0 0 0 2 4234 1 480 1 2750 0 0 5. CONCLUSION This paper discussed the optimal crashing time scheduling using for mega projects when the desired crashing time is less than the maximum possible crashing time . This is performed by linear programming technique in two phases. In the first stage, the mega project is analyzed to obtain the optimal amount of crashing time that should taken from the completion time of each critical milestone. in the second phase, these amounts are optimally distributed for crashing the critical activities of each milestone. For the considered example, it is assumed that the desired crashing time of the mega project is 106 days, which is less than . The optimal results assert to fully crash milestones , partially crash milestone , and leaving milestone uncrashed. Therefore, in the second phase, only milestone is optimally crashed. In this example, the total additional cost is 2.04% of the total project budget to achieve 108 days crashing of the mega project. Using this methodology, the manager can easily adjust the crash time-cost tradeoffs. However, the manager should realize that there may occur infeasible integer solutions with some desired crashing time. REFERENCES [1] PMBOK Guide, A Guide to the Project Management Body of Knowledge, 5th addition, Project Management Institute, Inc. 2013 [2] Brookes, Naomi J.; Locatelli, Giorgio (2015-10-01). Power plants as megaprojects: Using empirics to shape policy, planning, and construction management. Utilities Policy. 36: 57–66. doi:10.1016/j.jup.2015.09.005. [3] What You Should Know About Megaprojects and Why: An Overview. Project Management Journal. 7 Apr 2014. [4] Flyvbjerg et al., 2003; Locatelli et al., 2014a; Merrow, 2011; Van Wee, 2007. [5] Merrow, E.W., 2011. Industrial Megaprojects: Concepts, Strategies and Practices for Success. John Wiley & Sons. [6] Koch, C., 2012. Contested overruns and performance of offshore wind power plants. Constr. Manag. Econ. 30 (8), 609e622. [7] Van Marrewijk, Alfons, Stewart, R. Clegg, Tyrone, S. Pitsis, Marcel, Veenswijk, August 2008. Managing publiceprivate megaprojects: paradoxes, complexity, and project design. Int. J. Proj. Manag. 26 (6), 591e600. [8] Lima, M.B., Silva, L.B., and Vieira, R.J., 2006, Project Crashing and Costs Laws in the Knowledge Age, The 3rd International Conference on Production Research. http://www.iaeme.com/IJM/index.asp 121 editor@iaeme.com
Rami A. Maher, Ibrahim Abed Mohammad and Hamzeh H. Al-Safadi [9] Suri, P.K., and Bhushan, B., 2008, Simulator for Optimization of Software Project Cost and Schedule, Journal of Computer Science, vol. 12, pp. 1030-1035. [10] Omar M. Elmabrouk, Scheduling Project Crashing Time using Linear Programming Technique Proceedings of the 2012 International Conference on Industrial Engineering and Operations Management, Istanbul, Turkey, July 3 – 6, 2012 [11] Amer Mohammad Al-Qnahrah, Optimization of Multi-Resource Allocation in Large- Scale project Management, M.S. Thesis submitted to Isra University, 2015s [12] Floyd, Robert Warshall., Algorithm 97:Shortest Path, Communication of the ACM 5(6):345, 1962 [13] Rami A. Maher, Reem Aldouri, A Heuristic Algorithm of Resource Scheduling for Medium- Scale Project, First Isra Conference, PEMCON, 2015 http://www.iaeme.com/IJM/index.asp 122 editor@iaeme.com