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

ABSTRACT

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.

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

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

Cite this paper

nullR. 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.

nullR. 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,"

References

[1] M. L. Puri and D. A. Ralescu, “Differentials for Fuzzy Functions,” Journal of Mathematical Analysis and its Application, Vol. 91, No. 2, 1983, pp. 552-558.

[2] O. Kaleva, “The Cauchy Problem for Fuzzy Differential Equations,” Fuzzy Sets and Systems, Vol. 35, No. 3, 1990, pp. 389-396.

[3] C. X. Wu and M. Ma, “Embedding Problem of Fuzzy Number Space: Part I,” Fuzzy Sets and Systems, Vol. 44, No. 1, 1991, pp. 33-38.

[4] C. X. Wu, “An (α, β)-Optimal Solution Concept in Fuzzy Optimization Problems,” Optimization, Vol. 53, No. 2, 2004, pp. 203-221.

[5] C. X. Wu, “Evaluate Fuzzy Optimization Problems Based on Bi Objective Programming Problems,” Computer and Mathematics with Applications, Vol. 47, No. 5, 2004, pp. 893-902.

[6] M. Rabbani, M. A. Bajestani and G. B. Khoshkhou, “A Multi-Objective Particle Swarm Optimization for Project Selection Problem,” Expert Systems with Applications, Vol. 37, No. 1, 2010, pp. 315-321.

[7] M. Rabbani, R. T. Moghaddam, F. Jolai and H. R. Ghorbani, “A Comprehensive Model for R and D Project Portfolio Selection with Zero - One Linear Goal Programming,” IJE Transactions A: Basic, Vol. 19, No. 1, 2006, pp. 55-66.

[8] Y. Fang, L. Chen and M. Fukushima, “A Mixed R&D Projects and Securities Portfolio Selection Model,” European Journal of Operational Research, Vol. 185, No. 2, 2008, pp. 700-715.

[9] S. Riddell and W. A. Wallace, “The Use of Fuzzy Logic and Expert Judgment in the R & D Project Portfolio Selection Process,” Proceedings: PICMET, 2007, pp. 1228- 1238.

[10] H. Eilat, B. Golany and A. Shtub, “Constructing and Evaluating Balanced Portfolios of R&D Projects with Interactions: A DEA Based Methodology,” European Journal of Operational Research, Vol. 172, No. 3, 2006, pp. 1018-1039.

[11] C. Stummer and K. Heidenberger, “Interactive R & D Portfolio Selection Considers Multiple Objectives, Project Interdependencies, and Time: A Three Phase Approach,” Proceedings: PICMET, Portland, 2001, pp. 423- 428.

[12] J. D. Linton, S. T. Walsh, B. A. Kirchhoff, J. Morabito and M. Merges, “Selection of R&D Projects in a Portfolio,” Proceedings of IEEE Engineering Management Society, Washington, 2000, pp. 506-511.

[13] J. L. Ringuest, S. B. Graves and R. H. Case, “Conditional Stochastic Dominance in R & D Portfolio Selection,” IEEE Transactions on Engineering Management, Vol. 47, No. 4, 2000, pp. 478-484.

[14] R. L. Schmidt, “A model for R & D Project Selection with Combined Benefit, Outcome, and Resource Interactions,” IEEE Transactions on Engineering Management, Vol. 40, No. 4, 1993, pp. 403-410.

[15] O. Pereira and D. Junior, “The R & D Project Selection Problem with Fuzzy Coefficients,” Fuzzy Sets and Systems, Vol. 26, No. 3, 1988, pp. 299-316.

[16] M. A. Coffin and B. W. Taylor, “Multiple Criteria R & D Selection and Scheduling Using Fuzzy Logic,” Computers Operations Researches, Vol. 23, No. 3, 1996, pp. 207-220.

[17] L. L. Machacha and P. Bhattacharya, “A Fuzzy-Logic- Based Approach to Project Selection,” IEEE Transactions on Engineering Management, Vol. 47, No. 1, 2000, pp. 65-73.

[18] D. Kuchta, “A Fuzzy Mode for R & D Project Selection with Benefit: Outcome and Resource,” The Engineering Economist, Vol. 46, No. 3, 2001, pp. 164-180.

[19] S. Mohamed and A. K. McCowan, “Modelling Project Investment Decisions under Uncertainty Using Possibility Theory,” International Journal of Project Management, Vol. 19, No. 4, 2001, pp. 231-241.

[20] Y. G. Hsu, G. H. Tzeng and J. Z. Shyu, “Fuzzy Multiple Criteria Selection of Government-Sponsored Frontier Technology R & D Projects,” R & D Management, Vol. 33, No. 5, 2003, pp. 539-551.

[21] J. Wang and W. L. Hwang, “A Fuzzy Set Approach for R&D Portfolio Selection Using a Real Options Valuation Model,” Omega, Vol. 35, No. 3, 2007, pp. 247-257.

[22] S. S. Kim, Y. Choi, N. M. Thang, E. R. Ramos and W. J. Hwang, “Development of a Project Selection Method on Information System Using ANP and Fuzzy Logic,” World Academy of Science, Engineering and Technology, Vol. 53, No. 6, 2009, pp. 411-415.

[23] E. E. Karsak, “A Generalized Fuzzy Optimization Framework for R&D Project Selection Using Real Options Valuation,” Proceedings of ICCCSA, Berlin, Heidelberg, New York, 2006, pp. 918-927.

[24] B. Liu, “Uncertainty Theory,” 2nd Edition, Springer- Verlag, Berlin, 2007.

[25] B. Liu, “Fuzzy Process, Hybrid Process and Uncertain Process,” Journal of Uncertain Systems, Vol. 2, No. 1, 2008, pp. 3-16.

[26] B. Liu, “Some Research Problems in Uncertainty Theory,” Journal of Uncertain Systems, Vol. 3, No. 1, 2009, pp. 3-10.

[27] X. Li and B. Liu, “Hybrid Logic and Uncertain Logic,” Journal of Uncertain Systems, Vol. 3, No. 2, 2009, pp. 83-94.

[28] B. Liu, “Uncertainty Theory,” 3rd Edition. http://orsc.edu. cn/liu/ut.pdf

[29] X. Gao, “Some Properties of Continuous Uncertain Mea- sure,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 18, No. 4, 2007, pp. 383-390.

[30] C. You, “Some Convergence Theorems of Uncertain Sequences,” Mathematical and Computer Modelling, Vol. 49, No. 3-4, 2009, pp. 482-487.

[1] M. L. Puri and D. A. Ralescu, “Differentials for Fuzzy Functions,” Journal of Mathematical Analysis and its Application, Vol. 91, No. 2, 1983, pp. 552-558.

[2] O. Kaleva, “The Cauchy Problem for Fuzzy Differential Equations,” Fuzzy Sets and Systems, Vol. 35, No. 3, 1990, pp. 389-396.

[3] C. X. Wu and M. Ma, “Embedding Problem of Fuzzy Number Space: Part I,” Fuzzy Sets and Systems, Vol. 44, No. 1, 1991, pp. 33-38.

[4] C. X. Wu, “An (α, β)-Optimal Solution Concept in Fuzzy Optimization Problems,” Optimization, Vol. 53, No. 2, 2004, pp. 203-221.

[5] C. X. Wu, “Evaluate Fuzzy Optimization Problems Based on Bi Objective Programming Problems,” Computer and Mathematics with Applications, Vol. 47, No. 5, 2004, pp. 893-902.

[6] M. Rabbani, M. A. Bajestani and G. B. Khoshkhou, “A Multi-Objective Particle Swarm Optimization for Project Selection Problem,” Expert Systems with Applications, Vol. 37, No. 1, 2010, pp. 315-321.

[7] M. Rabbani, R. T. Moghaddam, F. Jolai and H. R. Ghorbani, “A Comprehensive Model for R and D Project Portfolio Selection with Zero - One Linear Goal Programming,” IJE Transactions A: Basic, Vol. 19, No. 1, 2006, pp. 55-66.

[8] Y. Fang, L. Chen and M. Fukushima, “A Mixed R&D Projects and Securities Portfolio Selection Model,” European Journal of Operational Research, Vol. 185, No. 2, 2008, pp. 700-715.

[9] S. Riddell and W. A. Wallace, “The Use of Fuzzy Logic and Expert Judgment in the R & D Project Portfolio Selection Process,” Proceedings: PICMET, 2007, pp. 1228- 1238.

[10] H. Eilat, B. Golany and A. Shtub, “Constructing and Evaluating Balanced Portfolios of R&D Projects with Interactions: A DEA Based Methodology,” European Journal of Operational Research, Vol. 172, No. 3, 2006, pp. 1018-1039.

[11] C. Stummer and K. Heidenberger, “Interactive R & D Portfolio Selection Considers Multiple Objectives, Project Interdependencies, and Time: A Three Phase Approach,” Proceedings: PICMET, Portland, 2001, pp. 423- 428.

[12] J. D. Linton, S. T. Walsh, B. A. Kirchhoff, J. Morabito and M. Merges, “Selection of R&D Projects in a Portfolio,” Proceedings of IEEE Engineering Management Society, Washington, 2000, pp. 506-511.

[13] J. L. Ringuest, S. B. Graves and R. H. Case, “Conditional Stochastic Dominance in R & D Portfolio Selection,” IEEE Transactions on Engineering Management, Vol. 47, No. 4, 2000, pp. 478-484.

[14] R. L. Schmidt, “A model for R & D Project Selection with Combined Benefit, Outcome, and Resource Interactions,” IEEE Transactions on Engineering Management, Vol. 40, No. 4, 1993, pp. 403-410.

[15] O. Pereira and D. Junior, “The R & D Project Selection Problem with Fuzzy Coefficients,” Fuzzy Sets and Systems, Vol. 26, No. 3, 1988, pp. 299-316.

[16] M. A. Coffin and B. W. Taylor, “Multiple Criteria R & D Selection and Scheduling Using Fuzzy Logic,” Computers Operations Researches, Vol. 23, No. 3, 1996, pp. 207-220.

[17] L. L. Machacha and P. Bhattacharya, “A Fuzzy-Logic- Based Approach to Project Selection,” IEEE Transactions on Engineering Management, Vol. 47, No. 1, 2000, pp. 65-73.

[18] D. Kuchta, “A Fuzzy Mode for R & D Project Selection with Benefit: Outcome and Resource,” The Engineering Economist, Vol. 46, No. 3, 2001, pp. 164-180.

[19] S. Mohamed and A. K. McCowan, “Modelling Project Investment Decisions under Uncertainty Using Possibility Theory,” International Journal of Project Management, Vol. 19, No. 4, 2001, pp. 231-241.

[20] Y. G. Hsu, G. H. Tzeng and J. Z. Shyu, “Fuzzy Multiple Criteria Selection of Government-Sponsored Frontier Technology R & D Projects,” R & D Management, Vol. 33, No. 5, 2003, pp. 539-551.

[21] J. Wang and W. L. Hwang, “A Fuzzy Set Approach for R&D Portfolio Selection Using a Real Options Valuation Model,” Omega, Vol. 35, No. 3, 2007, pp. 247-257.

[22] S. S. Kim, Y. Choi, N. M. Thang, E. R. Ramos and W. J. Hwang, “Development of a Project Selection Method on Information System Using ANP and Fuzzy Logic,” World Academy of Science, Engineering and Technology, Vol. 53, No. 6, 2009, pp. 411-415.

[23] E. E. Karsak, “A Generalized Fuzzy Optimization Framework for R&D Project Selection Using Real Options Valuation,” Proceedings of ICCCSA, Berlin, Heidelberg, New York, 2006, pp. 918-927.

[24] B. Liu, “Uncertainty Theory,” 2nd Edition, Springer- Verlag, Berlin, 2007.

[25] B. Liu, “Fuzzy Process, Hybrid Process and Uncertain Process,” Journal of Uncertain Systems, Vol. 2, No. 1, 2008, pp. 3-16.

[26] B. Liu, “Some Research Problems in Uncertainty Theory,” Journal of Uncertain Systems, Vol. 3, No. 1, 2009, pp. 3-10.

[27] X. Li and B. Liu, “Hybrid Logic and Uncertain Logic,” Journal of Uncertain Systems, Vol. 3, No. 2, 2009, pp. 83-94.

[28] B. Liu, “Uncertainty Theory,” 3rd Edition. http://orsc.edu. cn/liu/ut.pdf

[29] X. Gao, “Some Properties of Continuous Uncertain Mea- sure,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 18, No. 4, 2007, pp. 383-390.

[30] C. You, “Some Convergence Theorems of Uncertain Sequences,” Mathematical and Computer Modelling, Vol. 49, No. 3-4, 2009, pp. 482-487.