Uncertainty Theory Based Novel Multi-Objective Optimization Technique Using Embedding Theorem with Application to R & D Project Portfolio Selection ()

Rupak Bhattacharyya, Amitava Chatterjee, Samarjit Kar

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**DOI: **10.4236/am.2010.13023
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This paper introduces a novice solution methodology for multi-objective optimization problems having the coefficients in the form of uncertain variables. The embedding theorem, which establishes that the set of uncertain variables can be embedded into the Banach space C[0, 1] × C[0, 1] isometrically and isomorphically, is developed. Based on this embedding theorem, each objective with uncertain coefficients can be transformed into two objectives with crisp coefficients. The solution of the original m-objectives optimization problem with uncertain coefficients will be obtained by solving the corresponding 2 m-objectives crisp optimization problem. The R & D project portfolio decision deals with future events and opportunities, much of the information required to make portfolio decisions is uncertain. Here parameters like outcome, risk, and cost are considered as uncertain variables and an uncertain bi-objective optimization problem with some useful constraints is developed. The corresponding crisp tetra-objective optimization model is then developed by embedding theorem. The feasibility and effectiveness of the proposed method is verified by a real case study with the consideration that the uncertain variables are triangular in nature.

Keywords

Uncertainty Theory, Uncertain Variable, Embedding Theorem, α-Optimistic and α-Pessimistic Value, R & D Project Portfolio Selection

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R. Bhattacharyya, A. Chatterjee and S. Kar, "Uncertainty Theory Based Novel Multi-Objective Optimization Technique Using Embedding Theorem with Application to R & D Project Portfolio Selection," *Applied Mathematics*, Vol. 1 No. 3, 2010, pp. 189-199. doi: 10.4236/am.2010.13023.

Conflicts of Interest

The authors declare no conflicts of interest.

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