Abstract
This paper deals with the stochastic approach of bi-level multi-objective linear fractional programming problem.In this type of bi-level programming problem stochastic nature the right hand side resource vector is considered to follow a general form of distribution F (bi) = 1 − Bi^exp(Aih(bi))[13], which in itself includes many well known distributions such as Pareto distribution, Weibull distribution etc. After converting the problem into an equivalent deterministic form, each level of the problem is transformed into a single objective by using K-T conditions. Finally the problem is solved by Taylors series approach. A numerical example is also presented to illustrate how the proposed approach is utilized.
Highlights
Bi-level multi-objective programming problem (BLMOPP),which is an apparatus for modeling decentralized decisions, consists of the objectives of the first level decision maker (ULDM) at its first level and that of the objectives of the second level decision maker (LLDM) at the second level.The execution of decision is sequential, from first level to second level; each decision maker (DM) independently controls only a set of decision variables and optimizes the net benefits over a common feasible region
The objective functions of the DMs are linear fractional in the nature and the right hand side follows the general form of distributionsF = 1 − Bie−Aih(bi).Its deterministic equivalent form is derived for marginal constraints
Xn subject to ckj xj + γij dkj xj + θij for i=1, we have j = 1, 2, ...., m1, for upper level decision maker (ULDM) objective functions, for i=1,2,3....p,we have j = 1, 2, ...., mi, for LLDM objective functions, where mi, i = 1, 2, ...., p is the number of decision makers LLDM objective function, m is the number of constraint Ai is the coefficient of matrices of sizen m × ni, ckj, dkj ∈ Rn, dkj xj + θij > 0 for all x ∈ G and γij, θij are constants
Summary
Bi-level multi-objective programming problem (BLMOPP),which is an apparatus for modeling decentralized decisions, consists of the objectives of the first level decision maker (ULDM) at its first level and that of the objectives of the second level decision maker (LLDM) at the second level.The execution of decision is sequential, from first level to second level; each decision maker (DM) independently controls only a set of decision variables and optimizes the net benefits over a common feasible region. A stochastic version of the bi-level knapsack problem has given by Dempe and Richter in [15] In their variant of the problem the decision of the leader consists in choosing the (one dimensional) right hand side of the knapsack constraint (i.e. the capacity of the knapsack). Et al extend this model by introducing an uncertainty in the lower level problem They assume that the right hand side of the knapsack constraint in the lower level does depend on the leaders decision and on a random variable. The objective functions of the DMs are linear fractional in the nature and the right hand side follows the general form of distributionsF (bi) = 1 − Bie−Aih(bi).Its deterministic equivalent form is derived for marginal constraints. Results are illustrated with the help of numerical example
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