An algorithm to solve multi-objective fractional bottleneck transportation problem with restrictions

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An algorithm to solve multi-objective fractional bottleneck transportation problem with restrictions

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The transportation problems with multiple objectives having more than one restriction are very important from the practical point of view. In this paper, an algorithm is developed to solve Multiple Fractional Time Transportation Problem with Restrictions (MFTPR), which is solved by a lexicographic primal code. The developed algorithm is supported by a real life example of Metal Scrap Trade Corporation Limited. The multiple objective fractional transportation problem model development offers a more universal operation for a wider class of real-life transportation system decision problems than the fractional transportation problems because they take care of those real life planning problems from the economic world which have the mathematical structure of a transportation problem but are characterized by the existence of several fractional objective functions.

Keywords: Time Transportation, Fractional Programming, Restrictions, Lexicographic, Optimization, Multiple Objective

Abstract
Introduction
Mathematical formulation
Lexicographic multiple fractional time transportation problem with restrictions
Multiple objective fractional dual and optimality conditions
The MFTPR algorithm
Steel transportation problem of metal scrap trade corporation limited (MSTC)
Conclusion
References

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[...] f ijk k ' ~ ~ ~ = min ij ij [...]

[...] Let xij be the tonnage sent from i to j then it is required to a2 t ij min t 2 = max s 2 , j ) t ij a t ij 3 min t 3 = max s 3 , j ) t ij ) > ij ) > ij a1 t ij min t1 = max s1 , j ) t ij (xij ) > subject to j = ij = ai = 1,2,L i = ij = bj ( j = i ij p .x i = L jb j H j b j The Fractional Time Matrix T of the following related Lexicographic Multiple Fractional Time Transportation Problem with Restrictions (LMFTPR); r .x i i = ij x ij 0 = j = 1,2,L A lower bound for first objective is obtained as lex min ℑ = subject to xij i j β ij xij i j α ij j = ij = a i , = L 6 ) = b j , ( j = L 6 ) L jbj t la1 = max ( 480) = 780 t ls1 = max ( 480) = 780 The initial feasible basic solution MFTPR is i = i = ij X 0 to p .x i ij x14 = x16 = , x 22 = , x 23 = , x33 ri xij36≤=H j b jx 41 = x i = x 43 = ε , x54 = x64 = .x65 = x71 = 24, x73 = , x 74 = x75 = x 76 = , xij can be written as x82 = , x83 = 10, x84 = x85 = x86 = x11 = x12 = With the resulting bottleneck transportation time, the upper bounds are tU = 840 and s tU1 = 834. [...]

[...] By imposing conditions on ai and b j , one of the equations and is dependent and so a basic feasible solution will consist of NP+M+N-1 basic variables Lexicographic Multiple Fractional Time Transportation Problem with Restrictions The MFTPR, where fractional time objective function is to be minimized, is formulated as a Lexicographic Multiple Fractional Time Transportation Problem with Restrictions (LMFTPR) as β ij [ed ] j ξ dsf d = ( g + K h ) and (α ij , β ij ℜ h ) The above related LMFTPR can easily be solved by applying a lexicographic primal code. [...]

[...] Dutta Rao JR, Tiwari RN (1993) A restricted class of multi-objective linear fractional programming problems. European Journal of Operational Research 68: Mishra SK (2000) Multiobjective second order symmetric duality with cone constraints. European Journal of Operational Research 126:675- Mockus JB, Mockus LJ (1991) Bayesian approach to global optimization and application to multi-objective and constrained problems. Journal of Optimization Theory and Applications 70:157- Singh Saxena PK (1998) Total shipping cost/completion-date trade-offs in transportation problem with additional restrictions. Journal of Interdisciplinary Mathematics 1:161- [...]

[...] The algorithm is supported by a real life example of Metal Scrap Trade Corporation Limited (MSTC) Mathematical Formulation The mathematical formulation of the Multiple Fractional Time Transportation Problem with Restrictions (MFTPR) is M N t a1 ij = = x ij min t1 = max iM1 jN1 s1 , j ) tij i j M N a2 tij = = min t 2 = max iM1 jN x ij s2 , j ) tij i j = M M M N t aF ij = = min t F = max iM1 jN1 sF , j ) tij i j = subject to x ij j N ij = ai = 1,2,K M ) ( j = 1,2,K N ) i M i M ij = bj ijk xij jk ( j = 1,2,K N ; k = 1,2,K xij 0 = 1,2,K M ; j = 1,2,K N ) th th where ai is the amount of the commodity available at the i supply point and b j is the requirement of the commodity at the j demand point. [...]

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