A new proactive approach to construct a robust baseline schedule considering quality factor

Transcription

1 Int. J. Industrial and Systems Engineering, Vol. 22, No. 1, A new proactive approach to construct a robust baseline schedule considering quality factor Behrouz Afshar-Nadjafi Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, P.O. Box , Qazvin, Iran afsharnb@alum.sharif.edu Abstract: Extensive research has been devoted to project scheduling problems. However, little attention has been paid to quality factor. The assumption of deterministic parameters especially perfect output of accomplished activities is common in the project planning, whereas activities quality is subject to uncertainty. This may lead to rework for activities, as a consequence, serious revisions of the schedule baseline. In this paper, we consider the robust project scheduling problem to cope with reworks during project execution. A recursive procedure is proposed to solve the problem including time buffers between activities which is a proven method to improve the robustness of a baseline schedule. Also, an extensive simulation-based analysis of the performance of proposed procedure is given. Finally, the results are discussed in detail. Keywords: project scheduling; quality; robust; proactive; recursive; heuristic. Reference to this paper should be made as follows: Afshar-Nadjafi, B. (2016) A new proactive approach to construct a robust baseline schedule considering quality factor, Int. J. Industrial and Systems Engineering, Vol. 22, No. 1, pp Biographical notes: Behrouz Afshar-Nadjafi received his PhD degree in Industrial Engineering from Sharif University of Technology in 2008 and is currently an Assistant Professor and Graduate Programs Manager of Industrial Engineering at the Islamic Azad University, Qazvin Branch, in Iran. His research interests include: project scheduling, and inventory and production planning. More specifically, he is working on modelling and solution methods including exact procedures, meta-heuristic algorithms and artificial intelligence techniques regarding discrete optimisation problems in reality. 1 Introduction Project scheduling is the part of project management that involves the construction of a baseline schedule which specifies for each activity the precedence and resource feasible start times that will be used as a baseline for project execution. The majority of recent efforts in project scheduling concentrate on developing procedures to generate workable baseline schedules that optimise some particular objective(s). Also, all parameters of project are assumed deterministic (Yan et al., 2009; Montoya-Torres et al., 2010; Suresh et al., 2011; Khoshjahan et al., 2013; Rahimi et al., 2013; Afshar-Nadjafi, 2014a, 2014b). Kanagasabapathi et al. (2009) analysed the performance of the existing and newly Copyright 2016 Inderscience Enterprises Ltd.

2 64 B. Afshar-Nadjafi proposed scheduling rules in the static resource constrained multi project environment by considering performance measures involving mean tardiness and the maximum tardiness of projects. During project execution, however, a real-life project will never execute exactly as it was planned due to uncertainty. As a result of non-deterministic environment, a schedule that has been properly optimised for its particular objectives, but determined without consideration to disturbances that may occur, is likely to yield results significantly weaker than expected when used in practice. In real-life stochastic project settings, project schedules should include some solution robustness to cope with the uncertainties during project execution such that the realised project schedule, i.e., the list of actually realised activity start times during project execution, will not differ too much from the original baseline schedule. The main cause of the need for constructing stable baseline schedules is the fixed delivery dates required by suppliers or subcontractors and advance booking of key staff or equipment to guarantee their availability (Herroelen and Leus, 2004a). That is why, some recent research efforts have focused on the generation of robust project baseline schedules that are protected against possible disruptions that may occur during schedule execution. To provide an accurate estimate of the schedule robustness, they introduced several new robustness measures (Hazır et al., 2010; Khemakhem and Chtourou, 2013). These measures could help project managers in discriminating solutions having the same makespan to choose the most robust schedule. One possibility to maximise solution robustness is to include safety in the baseline schedule in order to absorb the anticipated disruptions as well as possible. This is called proactive scheduling (Van de Vonder et al., 2006). Buffering is a common practice in proactive scheduling. Its aim is to guarantee the completion of an activity on time. However, the methods of buffering are often different in positioning and sizing. Deblaere et al. (2011a) proposed a stochastic methodology for the determination of a project execution policy and a vector of predictive activity starting times with the objective of minimising a cost function that consists of the weighted expected activity starting time deviations and the penalties or bonuses associated with late or early project completion. Also, they propose and evaluate a number of dedicated exact reactive scheduling procedures as well as a tabu search heuristic for repairing a disrupted schedule, under the assumption that no activity can be started before its baseline starting time (Deblaere et al., 2011b). Time, cost, and quality are significant elements for judging the successes of projects. A projects success depends on how well these constraints are balanced (Atkinson, 1999). The quality of accomplished activities is one of the most important factors which is assumed deterministically perfect in past researches (Agarwal et al., 2011; Shi and Blomquist, 2012; Fang and Wang, 2012; Afshar-Nadjafi et al., 2013), while projects are not always completed as scheduled without reworking or modification. Sudden unexpected changes in construction technology, techniques, materials, or human resources can create budgetary and scheduling pressures that in turn may increase the possibility of failure (Zeng et al., 200). The literature on solution methods considering quality factor in project scheduling is scant. A crucial issue in maximising quality as an objective in project management is the use of a proper measure of project quality. Icmeli-Tukel and Rom (199), for example, base their measure on the assumption that the need to rework project activities causes delays and costs. In their view, quality is maximised by minimising both estimated rework times and costs. Tiwari et al. (2009) proposed approaches for scheduling projects with heterogeneous resources to meet time and quality objectives. Tareghian and Taheri

3 A new proactive approach to construct a robust baseline schedule 65 (2006) and Tareghian and Taheri (200) have treated quality as an important factor in tradeoff problems, claiming that overall project quality attained by project activities should be maximised within a given deadline and budget. Kim et al. (2012) proposed a mixed integer linear programming model and procedure that accounts for potential quality loss cost (PQLC) associated with rework or modifications that may occur due to excessive crashing activities. Therefore, the contribution of this paper is twofold: first, a new efficient recursive heuristic is developed for the project scheduling problem for generating stable project baseline schedules considering quality of activities execution. Following common practice in project scheduling (Herroelen and Leus, 2004a, 2005) we adopt a two-stage approach that first solves the project scheduling problem using any exact or heuristic solution procedure (Agarwal et al., 2011; Fang and Wang, 2012; Afshar-Nadjafi et al., 2013), then afterwards adds buffer to the initial schedule. Second, we will analyse in this paper the effectiveness of the proposed method based on simulation. The remainder of the paper is organised as follows: Section 2 describes the problem. Section 3 explains the steps of our algorithm to solve the problem. Performance results are represented in Section 4. Finally, Section 5 contains the conclusions. 2 Problem definition The classical resource constrained project scheduling problem (RCPSP) involves the scheduling of project activities without pre-emption on a set K of renewable resource types in order to minimise the project s makespan. In sequent, assume a project represented in AON format by a directed graph G = {N, A} where the set of nodes, N, represents activities and the set of arcs, A, represents finish-start precedence constraints with a time-lag of zero. The activities are numbered from the dummy start activity 1 to the dummy end activity n and are topologically ordered, i.e., each successor of an activity has a larger activity number than the activity itself. It is assumed that for execution quality of each activity i, there is L i out-of-control state(s). The duration of activity i, when executed at in-control state (with probability p i0 ), is d i0. However, with some probabilities p il, activity i, will be in out-of-control state l with duration d il, L i pil = 1 ; i. l= 0 Without loss of generality, it is assumed that duration of activity i in out-of-control state l, d il, is equal or larger than d i0, based on the assumption that the need to rework or modification project activities causes delays. That means, when an activity faced with quality failure, an extra time and effort is needed to reformation of activity s execution. This extra time depends on the amount of quality deviation of executed activity. Each activity i requires r ik units of renewable resource type k during each time unit of its execution. For each renewable resource k the availability a k is constant throughout the project horizon T. Let Ω(t) to be set of activities in progress in period t. A schedule S is defined by a vector of activity start times and is said to be feasible if all precedence and renewable resource constraints are satisfied. Vector S is defined as realised starting times of project activities during actual schedule execution. Every non-dummy activity i has a weight w i that denotes the marginal cost of deviating its realised starting time during execution from its predicted starting time in the baseline schedule (Van de Vonder et al.,

4 66 B. Afshar-Nadjafi 2008). The objective is to construct a stable and feasible baseline schedule for this NPhard problem (Leus and Herroelen, 2005) by minimising the robustness cost function. Using the above notation, proactive RCPSP under the minimum robustness cost function objective can be conceptually formulated as follows: n Min E(R) = wie si si (1) i= 1 s j si + di0 ( i, j ) A (2) rik ak t = 1,, T ; k = 1,, K (3) i Ω( t) sn T (4) s 0, Integer i N (5) i The objective in equation (1) is to minimise the weighted sum of the expected absolute deviations between the predicted starting times of the activities in the baseline and their realised starting times due to rework or modification of predecessors during actual schedule execution. The constraint set given in equation (2) imposes the finish-start precedence relations among the activities. Constraint set in equation (3) take care of the renewable resources limitations. Constraint in equation (4) guarantees that project deadline is not violated. Equation (5) specifies that the decision variables s i are non-negative integers. 3 Proposed algorithm In this section, we explain a recursive procedure for robust project scheduling formulated in Section 2. Also, an example is used to illustrate the proposed procedure. 3.1 Procedure The algorithm exploits the basic idea that the robustness cost function of a project can be minimised by first scheduling activities using any exact or heuristic solution procedure, followed by a recursive heuristic procedure which finds an activity for which adding one time-unit buffer in front of activity proves to be beneficial. If any buffering lead to violation project deadline, the algorithm stops and the start times of the project activities are reported. More specifically, step 1 of the procedure determines start time of activities using any exact or heuristic solution procedure considering d i0 as duration of activity i. We used the genetic algorithm (GA) of Afshar-Nadjafi et al. (2013) (by relaxing problem to single mode) and Demeulemeester and Herroelen (1992) to obtain the RCPSP initial feasible solution. The procedure developed by Demeulemeester and Herroelen (1992) is an exact branch and bound algorithm, while one developed by Afshar-Nadjafi et al. (2013) is a meta-heuristic algorithm which can obtain a satisfying near-optimal solution when an exact solution is not available. In step 2, an un-buffered schedule is the subject of a recursive search.

5 A new proactive approach to construct a robust baseline schedule 6 At each iteration, starting from the schedule created thus far (at first iteration, starting from the un-buffered schedule obtained in step 1) a priority c i is calculated for each activity i N according to relation (6): L i ci = wi pil dil di0 ; i (6) l= 0 This relation gives more priority to the activities which have more probability of quality failure and more rework time. Then, the activity i with the highest priority c i is selected for buffering. Ties are broken on the basis of (first) longest path from activity i to activity n assuming c i as activities duration in forward pass calculations (second) arbitrarily. If priority c i of the selected activity is zero algorithm stops, else new schedule is obtained by adding one time-unit buffer in front of the selected activity. The successors of activity i are shifted to right if forced so by binding precedence or resource constraints. By doing so, precedence and resource feasibility of the new schedule is guaranteed. If the project deadline is not violated the new obtained schedule is accepted for next iteration by replacing d i0 = d i0 + 1, else added buffer is removed and another unselected activity is selected for buffering (if any). If there is no unselected activity for buffering without violation the project deadline, the algorithm stops and the last start times vector is reported as robust baseline schedule. The pseudo-code of the algorithm is shown in Table 1. Table 1 The pseudo-code of the algorithm Step 1 Obtain an initial feasible schedule S using any exact or heuristic solution procedure Step 2 Set Iteration = 1 Step 3 Compute priority c i for each activity using relation (6) Step 4 Select an unselected activity i with the highest priority c i for buffering. If there is no unselected activity or if priority c i of the selected activity is zero go to step Step 5 Update the vector S by adding one time-unit buffer in front of the selected activity. Shift to right the successors of activity i if forced so by precedence or resource constraints. Step 6 If the project deadline is not violated, accept the new schedule, replace d i0 = d i0 + 1, set Iteration = Iteration +1, go to step 3, else, remove the added buffer and go to step 4 Step Stop the algorithm, report the last start times vector S as baseline schedule 3.2 Example In this section, we illustrate our method on a problem instance. The corresponding AON project network is shown in Figure 1. There are eight activities (and two dummy activities) and one renewable resource type with an availability of.

6 68 B. Afshar-Nadjafi Figure 1 Example project network The resource requirements, the probabilities p il which activity i may be at in-control state or out-of-control state(s) and related durations d il are given in Table 2. For example, resource requirement of activity 5 is 2. Also, with probability 0. this activity will be at in-control state with duration of 6. However, there are two possibilities for activity 5 to be out-of-control with probabilities 0.2 and 0.1, and durations and 9, respectively. For ease of representation, we assume the weights w i for all activities equal to 1. Table 2 Example parameters Activity r ik p i d i p il , , d il , , 14 0 The optimal un-buffered schedule for this example is presented in Figure 2 with a makespan of 22. Figure 2 Optimal un-buffered schedule to the problem example Resources used Time Considering a project deadline equal to 30 we obtained the buffered schedule of Figure 3 using the proposed algorithm after 19 iterations.

7 A new proactive approach to construct a robust baseline schedule 69 Figure 3 Robust baseline schedule obtained from proposed recursive algorithm (see online version for colours) Resources used Time 4 Performance analysis In this section, some computational experience with the proposed heuristic on Patterson benchmark set (Patterson, 1984) will be presented. This benchmark test set contains 110 single mode instances with up to 51 activities and up to 3 resources. For all of these problems, the optimal solutions are known (Demeulemeester and Herroelen, 1992). 4.1 Experimental data The famous 110 Patterson problems collected by Patterson (1984) are adapted to evaluate the performance of recursive algorithm. The w i in robustness cost function is randomly generated from the interval [1, 10]. One in-control state and three out-of-control states are assumed for each problem instance. Duration of each activity i in Patterson problems set are considered as d i0. Three random integers are generated from the interval [1, d i0 ]. Then, the numbers are sorted and added to d i0. Finally, the results are considered as durations of out-of-control states in increasing order, i.e., d i1 is assumed the d i0 + smallest number, d i2 the d i0 + second smallest and d i3 the d i0 + greatest. The values of p il for each activity i are generated as follows: First, three random numbers are generated from the interval [0, 0.2]. Then, the numbers are randomly assigned to probabilities of out-of-control states, i.e., p i1, p i2 and p i3. Finally, (1 p i1 p i2 p i2 ) is assumed as p i0. The project deadlines for the problem instances are generated as follows. After solving the different test instances with the work of Afshar-Nadjafi et al. (2013) to obtain the (near) minimal makespan, the project deadline was generated by multiplying the (near) minimal makespan by a number randomly generated from the interval [1, 2]. 4.2 Experimental results The proposed algorithm was coded in MATLAB.12.0 and executed on a personal computer with an Intel Core2Dou, 2.5 GHz processor and 3 GB memory. All 110 Patterson problems are solved with proposed procedure. In this paper the work of Afshar-Nadjafi et al. (2013) (by relaxing problem to single mode) and Demeulemeester

8 0 B. Afshar-Nadjafi and Herroelen (1992) is used to obtain the RCPSP initial feasible schedule S (step 1 of proposed algorithm). A simulation-based procedure is used to create realised start times vector S as follows: for every problem instance, a set of ten executions are simulated by drawing different actual activity durations from the set {d i0, d i1, d i2 and d i3 } with probabilities {p i0, p i1, p i2 and p i3 }, respectively. To do this, for each activity i, we generate a random number from the interval [0, 1]. If the selected number falls in the intervals [0, p i0 ], [p i0, p i0 + p i1 ], [p i0 + p i1, p i0 + p i1 + p i2 ] and [p i0 + p i1 + p i2, 1], actual duration of activity i will be d i0, d i1, d i2 and d i3, respectively. In doing so, we try 1,100 simulation runs. All 110 Patterson problem instances are solved using proposed algorithm with optimal RCPSP solution (Demeulemeester and Herroelen, 1992) and solution obtained by the GA of Afshar-Nadjafi et al. (2013), as initial schedule, separately. For all procedures, we compute the average robustness cost (E(R)) obtained from 1,100 simulated runs. The average robustness cost (E(R)) for simulation is computed based on first runs as S and other runs as S. Results are reported in Table 3. %NO denotes percentage of problem instances for which a certain procedure concludes the minimum robustness cost. %RD denotes percentage of relative deviation from simulation results. Table 3 Computational results for Patterson problems set Procedure %NO E(R) %RD Optimal-RCPSP (Demeulemeester and Herroelen, 1992) GA-RCPSP (Afshar-Nadjafi et al., 2013) Proposed recursive algorithm with initial optimal-rcpsp Proposed recursive algorithm with initial GA-RCPSP Simulation Table 3 reveals that proposed algorithm with optimal solution of RCPSP as initial schedule, has best performance measured by %NO and E(R). In 55.46% of problem instances (61 instances from 110 instances) proposed recursive algorithm with optimal solution of RCPSP as initial schedule has been best with lowest average of robustness cost (93.10). Also, from Table 3 it is clear that, proposed recursive algorithm with RCPSP solution obtained from GA as initial schedule, has been superior in 38.18% of problem instances (42 instances from 110 instances). Surprisingly, RCPSP solution obtained from GA succeeds in 6.36% of problem instances ( instances from 110 instances). The overall, proposed recursive algorithm in 93.64% of problems instances obtained robust schedules near to realised ones (weighed average 12.0% relative deviation from realised schedules). 5 Conclusions In this paper, in order to incorporate the quality factor, we proposed a new efficient recursive heuristic for robust RCPSP based on buffering. The algorithm contains two steps; step 1 of the procedure determines start time of activities using any exact or heuristic solution procedure. In step 2, an un-buffered schedule is the subject of a recursive search. At each iteration, according to the computed priorities one activity is selected for buffering and one time-unit buffer is added in front of activity without

9 A new proactive approach to construct a robust baseline schedule 1 violating precedence or resource constraints. Simulation-base studies showed that proposed algorithm is efficient. The results illustrated that the total cost when the robust solution is used is generally lower than the cost of reconfiguring the deterministic solution. More exactly, it was shown that the robust solutions obtained from proposed procedure are better protected against quality failures and constraint violations, including time and cost overruns. Also, a sensitivity analysis on starting solution of proposed procedure was done. The study of the impact of different initial schedules reveals that optimal initial solutions conclude some better results than near-optimal ones. In future, one can develop different ideas considering quality factor in project scheduling. The results of this paper can be beneficial for project managers that looking for the baseline schedules considering quality failures which may result disturbances in practice. References Afshar-Nadjafi, B. (2014a) A solution procedure for preemptive multi-mode project scheduling problem with mode changeability to resumption, Applied Computing and Informatics, in Press, doi: (accessed 6 March 2014). Afshar-Nadjafi, B. (2014b) Resource constrained project scheduling subject to due dates: preemption permitted with penalty, Advances in Operations Research, Vol. 2014, Article ID 50516, Doi: (accessed 16 April 2014). Afshar-Nadjafi, B., Rahimi, A. and Karimi, H. (2013) A genetic algorithm for mode identity and the resource constrained project scheduling problem, Scientia Iranica, Vol. 20, No. 3, pp Agarwal, A., Colak, S. and Erenguc, S. (2011) A neurogenetic approach for the resource constrained project scheduling problem, Computers & Operations Research, Vol. 38, No. 1, pp Atkinson, R. (1999) Project management: cost, time and quality, two best guesses and a phenomenon, its time to accept other success criteria, International Journal of Project Management, Vol. 1, No. 6, pp Deblaere, F., Demeulemeester, E. and Herroelen, W. (2011a) Proactive policies for the stochastic resource constrained project scheduling problem, European Journal of Operational Research, Vol. 214, No. 2, pp Deblaere, F., Demeulemeester, E. and Herroelen, W. (2011b) Reactive scheduling in the multi-mode RCPSP, Computers & Operations Research, Vol. 38, No. 1, pp Demeulemeester, E.L. and Herroelen, W.S. (1992) A branch-and-bound procedure for the multiple resource constrained project scheduling problem, Management Science, Vol. 38, No. 12, pp Fang, C. and Wang, L. (2012) An effective shuffled frog leaping algorithm for resource constrained project scheduling problem, Computers & Operations Research, Vol. 39, No. 5, pp Hazır, Ö., Haouari, M. and Erel, E. (2010) Robust scheduling and robustness measures for the discrete time/cost trade-off problem, European Journal of Operational Research, Vol. 20, No. 2, pp Herroelen, W. and Leus, R. (2004a) The construction of stable project baseline schedules, European Journal of Operational Research, Vol. 156, No. 3, pp Herroelen, W. and Leus, R. (2004b) Robust and reactive project scheduling: a review and classification of procedures, International Journal of Production Research, Vol. 42, No. 8, pp Herroelen, W. and Leus, R. (2005) Project scheduling under uncertainty survey and research potentials, European Journal of Operational Research, Vol. 165, No. 2, pp

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