Multi Objective Project Scheduling Under Resource Constraints Using Algorithm of Firefly

Transcription

1 Jurnal UMP Social Sciences and Technology Management Vol. 3, Issue. 1,2015 Multi Objective Project Scheduling Under Resource Constraints Using Algorithm of Firefly Saeed Yaghoubi, School of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran Meisam Jafari Eskandari, Department of Industrial Engineering, Payame Noor University, Tehran, Iran Meysam Farahmand Nazar Department of Industrial Engineering, Payame Noor University, Tehran, Iran Corresponding Author Abstract Products or services of any organization are developed in terms of project or operation way. For example, automotive companies produce as operative method and construction companies work in project method. Half a century has passed since emergence and development of project management techniques, and in that time, a large part of efforts for promoting the concepts of project management is done for project schedule models development. In this paper, project scheduling problem is solved with limited resources (RCPSP) modeling and meta-heuristic algorithm of Firefly worm and the results were compared with NSGA-II algorithm, the results show strong performance of this algorithm in solving proposed problems of RCPSP and multiobjective RCPSP. Key words: project scheduling, resource constraints, the algorithm of Firefly, NSGA II algorithm Introduction Products or services of any organization are developed in terms of project or operation way. For example, automotive companies produce as operative method and construction companies work in project method. Today's competitive world features cause that even organizations that reach their products (services) in operation method, also need to do numerous projects in their organizations. This is due to the competitive needs for defining new projects in the field of designing new products, cost reduction initiatives, and plans to change production lines and more. Half a century has passed since emergence and development of project management techniques, and in that time, a large part of efforts for promoting the concepts of project management is done for project schedule models development. Purpose of timing a project is to determine the time of doing different activities of the project during its implementation and it is related to decision-making process in which one or more targets are optimized. In this paper, project scheduling problem is solved with limited resources (RCPSP) modeling and with meta-heuristic algorithm of Firefly. Statement of the problem and literature Resource constrained project scheduling problem (RCPSP) consists of activities that should be taking into account the limitations of resources and also limitation of timing transposition, as project completion time will be minimal. This problem in concept of project scheduling is considered a standard concept that has attracted many researchers attention. So this problem is considered a fairly basic issue that for many practical applications can be used with restrictions, and, on the other side, the problem of RCPSP has been developed from various angles. In this study, we have tried to review some of these efforts in the field of modeling and also generalizations made in certain areas in this issue and main issues of them are described. On the issue of project scheduling it is tried that an optimized timing of activities provided, for this reason, considering target functions the modeling is done. Also a prerequisite relationship among activities is one of the main limitations of this issue. Project scheduling problem is generally based on three factors are divided into activities, resources and objective function. From the perspective of activities, project scheduling problem with different scenarios are based on prerequisite activities of one or multiple performance states. RCPSP is a particular combination optimization problem that is defined by multiple (V, 347

2 Multi Objective Project Scheduling p, E, R, B, b), where V is a set of activities, p is a vector of time, E is a set of transposition relationships, R is a set of available resources, B is a vector of the access to resources and b is a matrix of demands. While project completion time minimization is almost most well-known objections, there are other targets depended on different time. Goals based on delay time, delay and early time is very important. Delay time of Lj is activity j, deviation and completion time difference Cj from due time of dj, so Lj = Cj-dj. Lateness is similar to delay time, with the difference that it cannot be negative E m a x { 0, d C } T m a x { 0, C d } j j j. Early time Ej is in contrast to the previous definition, and we have. Koolish [1], Natasemboon and Randhava [2] and Vyana and Pynho [3] have considered weighted lateness minimization.natasmboon and Randhava [1] have suggested completion time minimization of all works, while Ram and colleagues [4] have considered minimizing of total weighted completion time. Similarly, Nazareh and colleagues [5] studied time average minimization duration. Note that total completion time minimization and completion time average minimization are equivalent of each other. Custer and colleagues [6], in their article, reviewed rescheduling of difference. Ranjbar and colleagues [7] considered objective function that in which we follow resources weighted lateness fine cost. In this model it is assumed that reviewed recycled resources, machinery and equipment s are very expensive that are used in some projects and consequently they are not fully available over time of project. In addition, objectives based on NPV for multimode RCPSP (Varma and colleagues [11], Valigora [12], Seifi and colleagues [13]) resource investment problem (Najafi and Niaki [14]), RCPSP with minimum and maximum interval time (Neumann and Zimmerman [15]) have also been investigated. Iysmily and Ram [18] used objective function NPV in a problem with continuous activity times and resource capacities based on time.khosh Jahan et al. Considered their model goal to minimize net current value of lateness- earliness penalty charges. These researchers imagine a specific delivery date for any activity. The approach presented in this article could be applicable in on-time production problems, where an activity after or even before determined date of that will include fine. Model developed in this study have two important criterions in project management, these two criteria are project completion time minimization and resulted benefits maximization using NPV formula. The model developed is resolved by meta-heuristic approach. This approach is used for solving multi-objective problems. Proposed problem RCPSP This model consists of two objective functions. An objective function is used to decrease beginning time of last activity and objective function is to seek to maximize revenues from project, this revenue is calculated as net present value. The variables used in this study are as follows: is zero and one variable. If ith activity at tth time with mth method will be finished value is one otherwise the value is zero T is index related to the beginning of an activity that has two states the earliest time of activity end and the latest time of activity end T` time period when the dual source is charged at the beginning of that period i Index related to an activity M States of doing activities that each state has its own time and resources dim Time of ith activity with mth method lth unrecyclable source, total number of sources are l lth unrecyclable source required for ith activity that is done with mth method ith operating cash flow α Discount rate DD Representing project deadline that is twice the critical path Above variables represent indices and variables used in the model. The model is shown as below. ) 1( j j j ) 2( ( ) 843

3 Jurnal UMP Social Sciences and Technology Management Vol. 3, Issue.1,2015 ) 3( ) 4( S.t ( ) ) 5( ) 6( ) 7( { } Next the model is defined. This model is a two-objective question. In above model, first objective function (1) is to seek minimization of end activity start time, the more this value is minimized, and the more total end project is minimized. The second objective function (2) is to maximize the present value of cash flows. Each project has cash flow over its life time, this cash flow included inputs and outputs. Inputs are revenues and outputs are costs that will be shown by negative mark. In a project in some of the activities that deliverable goods are existed, revenue is resulted and in most activities a cost will be spent for work. In this objective function it is tried to maximize present value of these cash flows.limitation (3) states that any activity should be implemented; this activity is done in earliest or the latest time and by one existed method.limitation (4) is related to transposition restrictions, time of beginning one activity plus its run time must be smaller than the start time after that.limitation (5) is related to non-renewable resources. The consumer resources to carry out activities are used in different ways and should be less than the total amount of the resource. Once the non-renewable source is assigned to entire project, then duration of project, this amount will not be resupplied.limitation (6) ensures that the project is completed before the deadline.last limitation is related to decision variable.as mentioned, this model cannot be solved with conventional methods because of a NP-Hard problem; therefore, the meta-heuristic method is proposed to solve that. In the next section in connection with solving method used in this research and how to set parameters will be discussed and numerical results obtained by solving this problem are offered. Solving proposed model As stated earlier, due to being NP-Hard, these classes of issues are used from meta-heuristic algorithms that in this study Firefly algorithm used that is not used so far in the literature to solve these problems and it is another innovations for this study. To facilitate, in Firefly algorithm following three rules are considered: All fireflies are intersexes and attracted to each other regardless of their gender. Absorption intensity is proportional to the brightness of the fireflies. Thus, for two fireflies the one that have less intensity light is moving toward the one with greater brightness. If none are brighter than the other, they move randomly toward each other. The objective function is determined by the brightness of firefly in destination. Based on these three rules, basic steps of Firefly algorithm can be summarized in next page pseudo code. Light intensity and charm There are two important issues in the algorithm of Firefly, changing intensity of light and formulation of charm. For simplicity, we can always assume that the attractiveness of a firefly is determined with the light that shone.in the simplest case the light intensity is proportional to second power of inverse rule. In this equation is the intensity of the light source. Also, if we assume the intensity of light absorbed by the environment ᵧ, light intensity can be calculated with the opposite relationship: 843

4 Multi Objective Project Scheduling Figure 1. Pseudo-code of Firefly algorithms In this equation is the intensity of the light source. To avoid the problem of oneness 5 at r = 0 in expression combination impact of second power of inverse rule and environmental absorption can be estimated by Gaussian formula: ( ) Since attractiveness of the Fireflies is perceived proportional to the light intensity by adjacent Firefly, the attractiveness of a firefly is defined by the following equation: In this equation this formula: ( ) ( ) is the intensity of light in r = 0. Also, distance between the two Fireflies is determined by Finally, J firefly movement toward more attractive firefly is determined by the following formula: ( ) In this equation second expression is attractiveness and the third expression is a random motion that parameter α in which is a random number in the range of 0 to 1 and parameter is uniform distribution. Also we consider parameter b equal to 1.In this study, we offer an example in the context of a network and then provide model data using meta-heuristic methods to solve problems and theirs results are compared with each other. To evaluate the performance of Firefly algorithms results of different problems solving are compared with results of solving sorting genetic algorithm. Statistical analysis is used for this comparison. Desired example to solve As noted in this study the problem is solved by evaluating the effectiveness of the model and choosing the right solution. In this study, some issues as test problems are in operations that are used to solve and evaluate effectiveness of different solution methods. In this study, one problem is selected from these sets in order to evaluate model effectiveness. Sample problems in this site are generated based on PSPLIB software. Sample questions on this site are generated based on software RanGen. This information is shown in the following table: 853

5 Jurnal UMP Social Sciences and Technology Management Vol. 3, Issue.1,2015 Post- requirement Table 1 - prerequisite relationships of sample issues-an example is a prerequisite Sample problem pre requisite Number of postrequirement Number of doing states Activity number Procedure and the resources required for each method is as following table. In this study, problem sample 8 is used to evaluate the performance of fireflies algorithm. Below problem is number one that is in the context for example, other problems are in the annex. 853

6 Multi Objective Project Scheduling Table 2. - Sample issues Activity METHOD Time R N Cash Flow

7 Jurnal UMP Social Sciences and Technology Management Vol. 3, Issue.1,2015 Other information issues are presented in the appendix. In this study, 8 problem cases were solved and its results were analyzed. Genetic algorithm parameter setting and results of solution Here you have to set entering parameters for algorithm NSGAII. Test designing and Taguchi method are used to set the parameters. Parameters of this algorithm are as follows: npop: the initial population size - Pc: probability of intersection -Pm: probability of mutation occurrence - Maxit: maximum number of frequencies Factors table is as follows: Table 3 Genetic algorithm factors Parameter Symbol Levels npop A Pc B Pm C Maxit D For factor 4 in three levels Taguchi table is as follows: At this stage, to help each of the 9 tests of Taguchi with specified settings solving will be done for any level of test, MID index value is calculated This indicator is defined as follows: The average distance ideal from the ideal point (MID): Using this indicator, near distance between Pareto solutions and the ideal point of those responses are calculated. The relationship is as follows: ( ) ( ) In the above expression and are ideal points of objective function. N is the number of Pareto solutions, and and are first and second objective function per reply i. The less MID index is, the algorithm has higher priority because of generating responses with lower distance from ideal point. Using software MINITAB following conclusions can be made: N / S rate values are as following table from software MINITAB: S/N rate table for levels Also average values of MINITAB software according to different levels is as follows: Levels average table Above values graph is as follows: 858

8 Multi Objective Project Scheduling Figure 2 average factors graph Figure 3 S/N rate graph for different factors In the above diagram any level that has a greater S/N is selected, in this case values used in meta-heuristic algorithm in this study are as follows: Table 4 Optimized factor levels Optimum factor value Symbol Factor 80 A npop 0.99 B Pc 0.2 C Pm 50 D Maxit Now using these values that are set in the algorithm we solve the model. The method for answering a vector is values of zero and one. Representing response in this study is a vector of numbers zero and one. This vector is the case, for example, has become one of the components of the vector. This process can be represented as follows: The operators used in this algorithm: In this study two operators of the composition and mutation are used as follows. In composition operator, two single-point and multi-point methods are used whose description is stated in chapter two. For mutation operator, simple mutation or flip & flop mutation are used. In this operator, randomly a number between 1 and nvar is selected then related element in the reply vector is reviewed that if its value is zero then it will turn into one, and if the value is zero then it turns into one. Initial reply production: Initial reply production in this study is random. randi function is used for this that if range is defined for it as [0, 1], in this case, the numbers produced are zero and one. The number of generated nvar numbers is the number of decision variables. Selection: The roulette wheel is used for selection in this algorithm whose description is shown in chapter two.the number of iterations, the number of initial population, crossover rate and mutation rate are also determined using parameter setting that is expressed as follows. Results are expressed as follows using MATLAB software. Pareto frontier is displayed in the chart below for two objective functions as it is shown the increase of one cause to decrease the other: 854

9 Jurnal UMP Social Sciences and Technology Management Vol. 3, Issue.1,2015 Figure 4 - Pareto frontier diagram Changes in the net present value at each iteration as follows: Figure 5 - The NPV for any iteration Firefly algorithm Procedure of this algorithm is fully described in section 3. In this study we are to evaluate the performance of the algorithm.the results of solving model are shown in the following charts: Figure 6 - NPV changes chart for each iteration of the Fireflies Pareto frontier using Firefly approach is as follows: Figure 7 - Pareto frontier in Firefly approach Analysis and comparison of two algorithms results In this section, we review and analyze the output results of the algorithm. The statistical analysis is used for this and indeed we examine the issue that the results of which one of algorithms has produced better values. Two-sided t-test is used for this, but before doing we need to ensure normality of results distribution function: NSGAII results in a significant level of 0.95 have normal distribution. Kolomograph- Smirnov test is used for this. The results of MINITAB software is as follows: 855

10 Multi Objective Project Scheduling Figure 8 - Normality test results of NSGAII data Results for Firefly algorithm outputs as follows: Figure 9 - The results of Fireflies data normality The significance level is more than 0.05 in both of them; therefore output results of both algorithms follow a normal distribution. Now we express the hypothesis. Here it is assumed that the average current value resulted from algorithm solution NSGAII is equal to Firefly algorithm. Mutual t test hypothesis is used for this. In fact, hypothesis test is as follows: The results of this test are as follows: Difference = mu (NSGAII) - mu (FA) Estimate for difference: % lower bound for difference: T-Test of difference = 0 (vs >): T-Value = 6.13 P-Value = DF = 73 The results of this test is lead to reject the null hypothesis, meaning that the average NPV resulted from NSGAII is greater than these values in the FA that reflects the efficiency of the algorithm in this problem. Hypothesis test in connection with the completion of the project as second objective function is as follows. It is expected that the average point time on the efficient frontier in algorithm NSGAII will be a smaller algorithm of FA. The results are as follows: Difference = mu (NSGAII) - mu (FA) Estimate for difference: % upper bound for difference: 0.79 T-Test of difference = 0 (vs <): T-Value = P-Value = DF = 96 Using data of a problem, we cannot conclude that algorithms are reviewed for various problems and their results are in the table below. Next, the results of this study are presented for problem 8: 853

11 Jurnal UMP Social Sciences and Technology Management Vol. 3, Issue.1,2015 P. n Average time of Pareto answer (ti me objective function) using an algorithm NSGAII Average NP V Pareto replies using algorithm NSGAII Table 5. Results of samples problems solving Average Average NP Runtime Run time of V Pareto - Pareto replies NSGA tim replies(tim using II e FA e objective algorithm function) FA using an algorithm F A Statistic al test P- Valu e Top algorith m NSGAII NSGAII FA FA NSGAII NSGAII NSGAII NSGAII Conclusion In most researches in this context that was expressed in detail in previous section, focus is on one special kind of resources, but in this study in addition to resources considered. Activities are considered multi states that doing them is possible in several methods. And also problem is considered as multi-objective. Therefore, we can find difference of this study with other ones, also in this study, two new multi-objective meta-initiative NSGAII and fireflies are used to select better algorithm in order to solve them that in chapter four, NSGAII superiority is shown. In this study, data are assumed determined that is it was supposed that time of activities is resources consumption as determined numbers. The first proposal that is offered in this study is to use modeling approaches under non-determined conditions. Using optimization approach based on modeling RCPSP problem Using the approach of random planning in RCPSP problem modeling References 1. KOLISCH, R Integrated scheduling, assembly area-and part-assignment for large-scale, make-to-order assemblies. International Journal of Production Economics, 64, NUDTASOMBOON, N. & RANDHAWA, S. U Resource-constrained project scheduling with renewable and non-renewable resources and time-resource tradeoffs. Computers & Industrial Engineering, 32, VIANA, A. & PINHO DE SOUSA, J Using metaheuristics in multiobjective resource constrained project scheduling. European Journal of Operational Research, 120, ROM, W. O., TUKEL, O. I. & MUSCATELLO, J. R MRP in a job shop environment using a resource constrained project scheduling model. Omega, 30, NAZARETH, T., VERMA, S., BHATTACHARYA, S. & BAGCHI, A The multiple resource constrained project scheduling problem: A breadth-first approach. European Journal of Operational Research, 112, KUSTER, J., JANNACH, D. & FRIEDRICH, G Applying local rescheduling in response to schedule disruptions. Annals of Operations Research, 180, RANJBAR, M., KHALILZADEH, M., KIANFAR, F. & ETMINANI, K An optimal procedure for minimizing total weighted resource tardiness penalty costs in the resource-constrained project scheduling problem. Computers & Industrial Engineering, 62, MIKA, M., WALIGORA, G. & WĘGLARZ, J Tabu search for multi-mode resource-constrained project scheduling with schedule-dependent setup times. European Journal of Operational Research, 187, PADMAN, R. & ZHU, D Knowledge integration using problem spaces: A study in resource-constrained project scheduling. Journal of Scheduling, 9, BRUCKER, P., DREXL, A., MÖHRING, R., NEUMANN, K. & PESCH, E Resource-constrained project scheduling: Notation, classification, models, and methods. European journal of operational research, 112, VARMA, V. A., UZSOY, R., PEKNY, J. & BLAU, G Lagrangian heuristics for scheduling new product development projects in the pharmaceutical industry. Journal of Heuristics, 13, WALIGÓRA, G Discrete continuous project scheduling with discounted cash flows A tabu search approach. Computers & Operations Research, 35, F " Using genetic algorithmsand simulated refrigeration to solve the scheduling problem with limited resources in a few fashion projects and cash flow were Tnzy ( Technical Report.")International Journalof Industrial Engineering and Production Management, 19 )4(. Winter 853

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