PROJECT SCHEDULING FOR NETWORK PROBLEMS USING JOB SEQUENCING TECHNIQUE

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1 Volume 114 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu PROJECT SCHEDULING FOR NETWORK PROBLEMS USING JOB SEQUENCING TECHNIQUE G. Sowmiya 1, K. Selvakumari 2 1 Research Scholar, Department of Mathematics Vels University, Chennai, Tamil Nadu, India sowmisekar.16@gmail.com 2 Assistant Professor, Department of Mathematics, Vels University, Chennai, Tamil Nadu, India selvafeb6@gmail.com Abstract The main role of the classical sequencing problem is to provide the optimal sequence of the jobs on machines. The problems related to sequencing are Routing problem network. The complicated real life situations are defined in terms of impressions which was overcame by Zadeh with a powerful tool of fuzzy data. By job sequencing technique fuzzy network problem can be easily solved [7]. In fuzzy project network [8], the duration of each activity is represented by a triangular fuzzy number [4]. The defuzzification formula for duration time can be converted into crisps number. This paper aims to find the shortest route of project of the network problems using Johnsons algorithm for job sequencing technique [7]. AMS Subject Classification: Key Words and Phrases: Fuzzy set, Triangular fuzzy numbers, Job sequencing technique, Johnson s sequencing algorithm, Optimal distance

2 1 Introduction Routing problem networks are related to sequencing, they have been receiving increasing attention such problems usually occur in the areas of transportations and communication. A network problem involves the determination of a route from source city1 to city 2 for there exist a number of alternative paths at various stages the journey. The cost of journey, which may be function of distance, time or money, is different for different routes and the problem is to find the minimum cost route. Theoretically, the procedure involves determination of the cost of all the possible routes and selecting the one with minimum cost. Network problem can be easily solved by job sequencing technique [7]. A network is defined as a set of points or nodes which are connected by lines or links. A way of going from one node to another is called a route or path. The problem of sequencing may have some restriction placed on it, such as time for each job on each machine, availability of resources (men, machine, material, and space) etc. In sequencing problem, the effectiveness can be measured in terms of minimized cost, maximized profits, and minimized elapsed times. We construct M M adjacency matrices between nodes [6]. Here M denotes the node of the network. Instead of time for each job for each machine we required the distance from one node to another node and calculated the optimal sequence which occur minimal total distance between source to destination. The project aims to present an analytical method for measuring the criticality in a fuzzy project network [8] [9], where the duration time of each activity is epresented by triangular fuzzy number [4]. We use a new defuzzification formula for triangular fuzzy number can be converted into crisp value of the triangular fuzzy number. From this, we found the optimized path of the network problems using N M sequencing problems by solving N 2 job sequencing technique [7]. 2 Preliminaries Definition 1. (Fuzzy Set)[3] Let X be a set. A fuzzy set à on X is defined to be a function à : X [0, 1] or µ à : X [0, 1] Equivalently, a fuzzy set à is defined to be the class of objects having the following representation à = {(x, µ à (x)) : x X)} 154

3 where µã : X [0, 1], is a function called the membership function of Ã. Definition 2. The fuzzy number à is a fuzzy set whose membership function µã(x) satisfies the following conditions: 1. µã(x) is piecewise continuous; 2. A fuzzy set à of the universe of discourse X is convex; 3. A fuzzy set of the universe of discourse X is called a normal fuzzy set if x i X, µã(x i ) = 1. Definition 3 (Triangular Fuzzy Number). [4] Let à = (a 1, a 2, a 3 ) be a triangular fuzzy number where a 1, a 2, a 3 and real numbers. The membership function µã(x) is, x a 1 a 2 a 1 for a 1 x a 2 1 for x = a 2 µã(x) = a 3 x a 3 a 2 for a 2 x a 3 0 otherwise Definition 4 (Ranking function). [5] A triangular fuzzy number, à = (a 1, a 2, a 3 ) a ranking function of triangular fuzzy number is defined by R(Ã) = a 1 + 2a 2 + a 3 4 (1) 3 Working rule Step 1 Apply ranking method for triangular fuzzy number, we convert the defuzzified value. Step 2 Constrain M M adjacency matrices. Nodes denoted by M. Step 3 Convert adjacency matrices split into sub matrices. Step 4 Apply Johnson s method for n-jobs, 2-machine algorithm. Step 5 Find the shortest path of the project network from optimal sequence. 155

4 4 Numerical example Figure 1: Project network (Construction project network) Node 1: student -1 (surveyor); Node 2: student-2 (maintenance & safety engineer); Node 3: student-3 (executive engineer & quality engineer); Node 4: student-4 (store keeper); Node 5: student-5 (planning). Find the shortest optimal path for construction project network. Solution: Step 1 Find the defuzzified value using equation (1). Node (i j) Fuzzy capacities Defuzzified value 1-2 (4, 8, 16) (2, 4, 6) (2, 3, 8) (4, 6, 8) (2, 4, 10) (2, 7, 8) (4, 8, 20) (1, 3, 5) 3 156

5 Figure 2: Project network Step 2 The Adjacency matrix is, Node Step 3 The adjacency matrix split into sub matrix is, Node M 1 M 5 Optimal Sequence Node M 1 + M 2 M 4 + M 5 Optimal Sequence Node M 1 + M 2 + M 3 M 3 + M 4 + M 5 Optimal Sequence

6 Node M 1 + M 2 + M 3 + M 4 M 2 + M 3 + M 4 + M 5 Optimal Sequence Step 4 The total time for each sub matrix is, Sequence/Node Similarly we can find sequence of submatrix =45, =40, =41. From all the total time of sub matrices 35 is minimum Therefore the optimal path is construction project network. Figure 3: Optimal path (Sequence: = 35) 5 Conclusion A new analytical method for finding critical path in a fuzzy project network has been proposed. The optimized path of the network problems can be using the job sequencing technique. The technique is only used for machine means hardware for sequencing the jobs technique is used in networking problem as shortest route in finding a problem. We conclude that the method projected in this paper is an alternative way to identity the fuzzy project network. 158

7 References [1] H.N. Ahuja, S.P. Dozzi, S.M. Abourizk, Project Management, New York, Wiley (1994). [2] M. Blue, B. Bush, J. Puckett, Unified Approach to Fuzzy Graph Problems, Fuzzy Set and Systems, 125 (2002), [3] S. Chanas, P. Zielinski, Critical path analysis in the network with fuzzy task times, Fuzzy sets and systems, 122 (2001), [4] S. Elizabeth, L. Sujatha, Project Scheduling method using Triangular Intuitionistic fuzzy numbers and Triangular fuzzy numbers, Applied Mathematical Sciences, 9, No. 4 (2015), [5] K. Kripa, R. Govindarajan, Fuzzy Sequencing Problem, Internation journal of scientific Research, 5, Issue 5, Issn No [IF:3.508]IC Value: [6] M. Nawaz, A Heuristic Algorithm for n-job, m-machine Flow Shop Sequencing Problem, Omega, 11 (1983), [7] Punit kumar singh, Dr. Rakesh kumar, Path Optimization Algorithm For Network Problem Using Job Sequencing Technique, International Journal Of Distributed And Parallel System (IJDPS), 3, No. 3 (2012), [8] N. Ravi Shankar, V. Sireesha, P. Phani Bushan Rao, Critical path analysis in the fuzzy project network, Advances in fuzzy mathematics, 5, No. 3 (2010), , ISSN X. [9] N. Ravi Shankar, V. Sireesha, P. Phani Bushan Rao, An Analytical Method For Finding Critical Path in a Fuzzy Project Network, Int. J. Contemp. Math. Science, 5, No. 20 (2010),