Some Solid Transportation Models with Crisp and Rough Costs
In this paper, some practical solid transportation models are formulated considering per trip capacity of each type of conveyances with crisp and rough unit transportation costs. This is applicable for the system in which full vehicles, e.g. trucks, rail coaches are to be booked for transportation of products so that transportation cost is determined on the full of the conveyances. The models with unit transportation costs as rough variables are transformed into deterministic forms using rough chance constrained programming with the help of trust measure. Numerical examples are provided to illustrate the proposed models in crisp environment as well as with unit transportation costs as rough variables.
[1] K. Dembczynski, S. Greco, R. Slowinski, Rough set approach to multiple
criteria classification with imprecise evaluations and assignments,
European Journal of Operational Research 198 (2009) 626-636.
[2] S. Greco, B. Matarazzo, R. Slowinski, Rough sets theory for multicriteria
decision analysis, European Journal of Operational Research 129 (2001)
1-47.
[3] K.B.Haley, The sold transportation problem, Operations Research 10
(1962) 448-463.
[4] S. Hirano, S. Tsumoto, Rough representation of a region of interest
in medical images, International Journal of Approximate Reasoning 40
(2005) 23-34.
[5] F.Jim'enez, J.L.Verdegay, Uncertain solid transportation problems, Fuzzy
Sets and Systems, 100 (1998) 45-57.
[6] T. Lin, Y. Yao, L. Zadeh, Data Mining, Rough Sets, and Granular
Computing, Springer Verlag, 2002.
[7] P. Lingras, Rough Neural Networks, International Conference on Information
Processing and Management of Uncertainty, Granada, Spain, 1996,
1445-1450.
[8] B. Liu, Theory and Practice of Uncertain Programming, Physical-Verlag,
Heidelberg, 2002.
[9] B. Liu, Uncertainty Theory: An Introduction to its Axiomatic Foundation,
Springer-Verlag, Berlin, 2004.
[10] B. Liu, Inequalities and Convergence Concepts of Fuzzy and Rough
Variables, Fuzzy Optimization and Decision Making 2 (2003) 87-100.
[11] G. Liu, W. Zhu, The algebraic structures of generalized rough set theory,
Information Sciences 178 (2008) 4105-4113.
[12] S. Liu, Fuzzy total transportation cost measures for fuzzy solid transportation
problem, Applied Mathematics and Computation 174 (2006)
927-941.
[13] A. Nagarjan, K. Jeyaraman, Solution of chance constrained programming
problem for multi-objective interval solid transportation problem
under stochastic environment using fuzzy approach, International Journal
of Computer Applications 10(9) (2010) 19-29.
[14] A.Ojha, B.Das, S.Mondal, M.Maiti, An entropy based solid transportation
problem for general fuzzy costs and time with fuzzy equality,
Mathematical and Computer Modeling, 50 (1-2) (2009) 166-178.
[15] A. Ojha, B. Das, S. Mondal, M. Maiti, A stochastic discounted multiobjective
solid transportation problem for breakable items using analytical
hierarchy process, Applied Mathematical Modelling 34 (2010) 2256-
2271.
[16] Z. Pawlak, Rough sets, International Journal of Information and Computer
Sciences 11(5) (1982) 341-356.
[17] Z. Pawlak, Rough Sets - Theoretical Aspects of Reasoning About Data,
Kluwer Academatic Publishers, Boston, 1991.
[18] Z. Pawlak, R. Slowinski, Rough set approach to multi-attribute decision
analysis (invited review), European Journal of Operational Research 72
(1994) 443-459.
[19] Z. Pawlak, A. Skowron, Rough sets: some extensions, Information
Sciences 177 (2007) 28-40.
[20] L. Polkowski: Rough Sets, Mathematical Foundations, Advances in Soft
Computing, Physica Verlag, A Springer-Verlag Company, 2002.
[21] E.D.Schell, Distribution of a product by several properties, Proceedings
of 2nd Symposium in Linear Programming, DCS/comptroller, HQ US
Air Force, Washington D C (1955) 615-642.
[22] M. Shafiee, N. Shams-e-alam, Supply Chain Performance Evaluation
With Rough Data Envelopment Analysis, 2010 International Conference
on Business and Economics Research, vol.1 (2011), IACSIT Press, Kuala
Lumpur, Malaysia.
[23] Z. Tao, J. Xu, A class of rough multiple objective programming and
its application to solid transportation problem, Information Sciences 188
(2012) 215235.
[24] S. Tsumoto, Medical differential diagnosis from the viewpoint of rough
sets, Information Sciences (2007) 28-34.
[25] J. Xu and L. Yao, A Class of Two-Person Zero-Sum Matrix Games with
Rough Payoffs, International Journal of Mathematics and Mathematical
Sciences 2010, doi:10.1155/2010/404792.
[26] J. Xu, B. Li, D. Wu, Rough data envelopment analysis and its application
to supply chain performance evaluation, International Journal of
Production Economics 122 (2009) 628-638.
[27] Shu Xiao and Edmund M-K. Lai, A Rough programming approach to
power-aware VLIW instruction scheduling for digital signal processors,
ICASSP 2005.
[28] L. Yang, Y. Feng, A bicriteria solid transportation problem with fixed
charge under stochastic environment, Applied Mathematical Modelling
31 (2007) 2668-2683.
[29] L. Yang, L. Liu, Fuzzy fixed charge solid transportation problem and
algorithm, Applied Soft Computing 7 (2007) 879-889.
[30] L. Yang, Z. Gao, K. Li, Railway freight transportation planning with
mixed uncertainty of randomness and fuzziness, Applied Soft Computing
11 (2011) 778-792.
[31] E. Youness, Characterizing solutions of rough programming problems,
European Journal of Operational Research 168 (2006) 1019-1029.
[32] W. Zhu, Relationship between generalized rough sets based on binary
relation and covering, Information Sciences 179 (2009) 210-225.
[1] K. Dembczynski, S. Greco, R. Slowinski, Rough set approach to multiple
criteria classification with imprecise evaluations and assignments,
European Journal of Operational Research 198 (2009) 626-636.
[2] S. Greco, B. Matarazzo, R. Slowinski, Rough sets theory for multicriteria
decision analysis, European Journal of Operational Research 129 (2001)
1-47.
[3] K.B.Haley, The sold transportation problem, Operations Research 10
(1962) 448-463.
[4] S. Hirano, S. Tsumoto, Rough representation of a region of interest
in medical images, International Journal of Approximate Reasoning 40
(2005) 23-34.
[5] F.Jim'enez, J.L.Verdegay, Uncertain solid transportation problems, Fuzzy
Sets and Systems, 100 (1998) 45-57.
[6] T. Lin, Y. Yao, L. Zadeh, Data Mining, Rough Sets, and Granular
Computing, Springer Verlag, 2002.
[7] P. Lingras, Rough Neural Networks, International Conference on Information
Processing and Management of Uncertainty, Granada, Spain, 1996,
1445-1450.
[8] B. Liu, Theory and Practice of Uncertain Programming, Physical-Verlag,
Heidelberg, 2002.
[9] B. Liu, Uncertainty Theory: An Introduction to its Axiomatic Foundation,
Springer-Verlag, Berlin, 2004.
[10] B. Liu, Inequalities and Convergence Concepts of Fuzzy and Rough
Variables, Fuzzy Optimization and Decision Making 2 (2003) 87-100.
[11] G. Liu, W. Zhu, The algebraic structures of generalized rough set theory,
Information Sciences 178 (2008) 4105-4113.
[12] S. Liu, Fuzzy total transportation cost measures for fuzzy solid transportation
problem, Applied Mathematics and Computation 174 (2006)
927-941.
[13] A. Nagarjan, K. Jeyaraman, Solution of chance constrained programming
problem for multi-objective interval solid transportation problem
under stochastic environment using fuzzy approach, International Journal
of Computer Applications 10(9) (2010) 19-29.
[14] A.Ojha, B.Das, S.Mondal, M.Maiti, An entropy based solid transportation
problem for general fuzzy costs and time with fuzzy equality,
Mathematical and Computer Modeling, 50 (1-2) (2009) 166-178.
[15] A. Ojha, B. Das, S. Mondal, M. Maiti, A stochastic discounted multiobjective
solid transportation problem for breakable items using analytical
hierarchy process, Applied Mathematical Modelling 34 (2010) 2256-
2271.
[16] Z. Pawlak, Rough sets, International Journal of Information and Computer
Sciences 11(5) (1982) 341-356.
[17] Z. Pawlak, Rough Sets - Theoretical Aspects of Reasoning About Data,
Kluwer Academatic Publishers, Boston, 1991.
[18] Z. Pawlak, R. Slowinski, Rough set approach to multi-attribute decision
analysis (invited review), European Journal of Operational Research 72
(1994) 443-459.
[19] Z. Pawlak, A. Skowron, Rough sets: some extensions, Information
Sciences 177 (2007) 28-40.
[20] L. Polkowski: Rough Sets, Mathematical Foundations, Advances in Soft
Computing, Physica Verlag, A Springer-Verlag Company, 2002.
[21] E.D.Schell, Distribution of a product by several properties, Proceedings
of 2nd Symposium in Linear Programming, DCS/comptroller, HQ US
Air Force, Washington D C (1955) 615-642.
[22] M. Shafiee, N. Shams-e-alam, Supply Chain Performance Evaluation
With Rough Data Envelopment Analysis, 2010 International Conference
on Business and Economics Research, vol.1 (2011), IACSIT Press, Kuala
Lumpur, Malaysia.
[23] Z. Tao, J. Xu, A class of rough multiple objective programming and
its application to solid transportation problem, Information Sciences 188
(2012) 215235.
[24] S. Tsumoto, Medical differential diagnosis from the viewpoint of rough
sets, Information Sciences (2007) 28-34.
[25] J. Xu and L. Yao, A Class of Two-Person Zero-Sum Matrix Games with
Rough Payoffs, International Journal of Mathematics and Mathematical
Sciences 2010, doi:10.1155/2010/404792.
[26] J. Xu, B. Li, D. Wu, Rough data envelopment analysis and its application
to supply chain performance evaluation, International Journal of
Production Economics 122 (2009) 628-638.
[27] Shu Xiao and Edmund M-K. Lai, A Rough programming approach to
power-aware VLIW instruction scheduling for digital signal processors,
ICASSP 2005.
[28] L. Yang, Y. Feng, A bicriteria solid transportation problem with fixed
charge under stochastic environment, Applied Mathematical Modelling
31 (2007) 2668-2683.
[29] L. Yang, L. Liu, Fuzzy fixed charge solid transportation problem and
algorithm, Applied Soft Computing 7 (2007) 879-889.
[30] L. Yang, Z. Gao, K. Li, Railway freight transportation planning with
mixed uncertainty of randomness and fuzziness, Applied Soft Computing
11 (2011) 778-792.
[31] E. Youness, Characterizing solutions of rough programming problems,
European Journal of Operational Research 168 (2006) 1019-1029.
[32] W. Zhu, Relationship between generalized rough sets based on binary
relation and covering, Information Sciences 179 (2009) 210-225.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:53986", author = "Pradip Kundu and Samarjit Kar and Manoranjan Maiti", title = "Some Solid Transportation Models with Crisp and Rough Costs", abstract = "In this paper, some practical solid transportation models are formulated considering per trip capacity of each type of conveyances with crisp and rough unit transportation costs. This is applicable for the system in which full vehicles, e.g. trucks, rail coaches are to be booked for transportation of products so that transportation cost is determined on the full of the conveyances. The models with unit transportation costs as rough variables are transformed into deterministic forms using rough chance constrained programming with the help of trust measure. Numerical examples are provided to illustrate the proposed models in crisp environment as well as with unit transportation costs as rough variables.
", keywords = "Solid transportation problem, Rough set, Rough variable,
Trust measure.", volume = "7", number = "1", pages = "24-8", }
