An Application of Approximation Algorithm for Project Scheduling Problem based on Semi Definite Optimization Authored by T.Bakshi, A.SinhaRay, B.Sarkar, S.K.Sanyal Presented by Tuli Bakshi

 This is a project scheduling algorithm for better estimation of project in an incomplete information environment.  This approximation algorithm is one of the tools in theory and practice of the project scheduling problem.

Introduction In this paper, We have used the Goemans- Williamson’s max cut algorithm in a different way in the area of project scheduling problem. The framework described here is based on semi definite programming.

NP-COMPLETENESS OF SCHEDULING ALGORITHM An algorithm A is polynomial time algorithm if the input size is the polynomial function of x; otherwise it is an exponential time algorithm. A problem is tractable if there is a polynomial time algorithm; otherwise intractable. The theory of NP-hardness suggests that there is a large class of problems, namely, NP-hard problems that may be intractable.

NP-COMPLETENESS OF SCHEDULING ALGORITHM For scheduling problems, we considered that the number of jobs n and the number of machines m, should be the part of inputs. Precedence constraint poses no problem, since there can be at most O (n 2 ) precedence relations for n jobs. There are scheduling problems that are NP- hard with respect to binary encoding of inputs such as processing time, due dates, weights etc.

APPROXIMATING SCHEDULING ALGORITHM Approximation algorithm has been adopted to solve the problem which has high order of polynomial time or the problems belong to the NP-class. Approximation algorithm does not suffer from huge constants associated with time complexity bounds.

SDP as an Approximation Algorithm In order to claim that a semi-definite program is approximately solvable in polynomial time, we need to assume that it is “Well behaved” in some sense. Namely we need that the feasible solutions cannot be too large.

PROBLEM FORMULATION 1. Semi definite problem formulation and relaxation. 2. SDP Max Cut problem formulation. 3. GW Max Cut problem formulation. 4. Semi definite relaxation for Max Cut problem. 5. Relaxation to the above formulation: We have identified the following formulations and relaxations:

COMPUTATIONAL EXPERIMENT & RESULT In this section we compute and compare the results provided by deterministic activity, duration and finish-start precedence relation problem of the critical path calculation with semi-definite optimization application model proposed above.

Example

The value of the critical path 1 → 5 → 6 → 7 → 9 → 10 → 11. So the length is 125.

Solution by GW-MaxCut Algorithm

R ESULT OF MAX-CUT

So we choose (1) k = {1, 2, 4, 10} ∂(k) = { (1,5),(2,3),(4,7),(4,10),(10,11)} w∂ (k) = = 84.

C OMPLEXITY OF P ROPOSED A LGORITHM The proposed approximation algorithm achieves an upper bound with an error limit

CONCLUSION & FUTURE SCOPES OF PROPOSED ALGORITHM It can proved that the time complexity of critical path algorithm in worst-case O(n 2 ). The proposed approach described in this paper bears well than the well known CPM. The computational running time of the SDP based project scheduling algorithm is lesser than that of CPM.

◦ Our proposed algorithm is in the developing stage. We have tested this in various project scheduling problems. The most advantageous factor of this algorithm is that it can be applied to calculate the project duration in an condition i. e. where all the searches are not in their best possible cases, therefore this algorithm has a better tolerance property than that of CPM.

LIMITATIONS The main limitation of this proposed algorithm is that the SDP constraints are proportional to number of paths. Therefore, for a huge network, proper heuristic to be developed and that could be able to reduce the growing number of SDP constraints.

R EFERENCES S.E.Elmaghraby, “Project planning and control by network models”, John Wiley and Sons, J.H.Hagstrom, “Computational complexity of PERT problems”, Networks 18: , F.K. Levy and J.D. Wiest, “ A Management guide to PERT / CPM: with GERT/PDM/DPCM and other networks”,Prentice Hall 2 nd ed, N.Slack, S. Chambers, C. Harland, A. Harrison and R. Johnston, Operations Management, Pitman Publishing, Graham, R.L.,E.L. Lawler,J.K.Lenstra and A.H.G.Rinnooy Kan, “Optimization and Approximation in Deterministic Sequencing and Scheduling Theory: A Survey”, A Survey, Annals of Discrete Mathematics. Tien-Fu Liang, “Applying fuzzy goal programming to project management decisions with multiple goals in uncertain environments “, Expert Systems with Applications 37, ,2010. D. Bertsimas, K. Natarajan, C.P.Teo, “ Applications of Semidefinite Optimization in Stochastic Project Scheduling”, Massachusetts Institute of Technology and National University of Singapore.

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