BILEVEL TRANSPORTATION PROBLEMS

. In this paper we formulate the bilevel transportation problem of the cost-time type and of the cost-cost type, we propose a general algorithm for solving this problems and we also give two examples. MSC 2000. 90C08, 90C29.


INTRODUCTION
The bilevel (or two level) mathematical programming problem is an optimization problem with special constraints determined by, all or in part, another optimization problem.The bilevel mathematical programming problem is defined conventionally as follows: Find x * ∈ R n such that x * solves min F (x, y(x)) subject to G(x, y(x)) ≤ 0, where y(x) solves, for fixed x, min f (x, y) subject to g(x, y) ≤ 0, where G : R q × R p → R r , F : R q × R p → R, g : R q × R p → R s , and f : R q × R p → R, are functions.

BILEVEL TRANSPORTATION PROBLEM OF THE COST-TIME TYPE
Let a 1 , . . ., a m , b 1 , . . ., b n be positive integers so that a 1 + . . .+ a m = b 1 + . . .+ b n .Let M = {1, . . ., m}, P = {1, . . ., p}, S = {p + 1, . . ., n}, The bilevel transportation problem of the cost-time type is defined as follows: In the following we denote by C the matrix of costs and by T the matrix of times:

If we denote by
Let be Let the parametric transportation problem of the time type be where α = (α 1 , ..., α m ) ∈ H.If α 0 ∈ H , then by (P T P (α 0 )) we denote the transportation problem of the time type which is obtained from (PTP) for α = α 0 .
Solving the problem (P T P ) , we obtain a split of the set H into a finite number of subsets H 1 , . . ., H q ⊆ N m , such that H 1 ∪ • • • ∪ H q = H, and for each k ∈ {1, . . ., q}, there are a real number T k , with the property and a matrix, which we denote by Y k (α), is an optimal solution for the problem (P T P (α)) for each α ∈ H k .Then a solution of the problem (P T P ) is a function h : H → N m×(n−p) , given by for each k ∈ {1, ..., q}.
For each k ∈ {1, . . ., q}, we solve the parametric transportation problem of the cost type where α ∈ H k .
For each α ∈ H k we denote by X k (α) an optimal solution of the problem (T P k (α)), obtained from (T P k ) when α is fixed. Let Then, we call the matrix X k = X k (α * ), the best solution of the problem (T P k ).The solution of the bilevel transportation problem (BTP) is that X k for which we have In order to illustrate the above algorithm we conclude with the following numerical example.
Hence the optimal solution for the bilevel transportation problem of the cost-time type is The bilevel transportation problem of the cost-cost type is defined as follows: In the following we denote by C and D the matrices of costs: Let be Let the parametric transportation problem of the cost type be where α = (α 1 , ..., α m ) ∈ H.If α 0 ∈ H , then by (P CP (α 0 )) we denote the transportation problem of the cost type which is obtained from (PCP) for α = α 0 .
Solving the problem (P CP ) , we obtain a split of the set H into a finite number of subsets H 1 , . . ., H q ⊆ N m , such that H 1 ∪ • • • ∪ H q = H, and for each k ∈ {1, . . ., q}, there are a real number d k , with the property g * (α) = d k for all α ∈ H k , and a matrix, which we denote by Y k (α), for each k ∈ {1, ..., q}.
For each k ∈ {1, . . ., q}, we solve the parametric transportation problem of the cost type where α ∈ H k .
For each α ∈ H k we denote by X k (α) an optimal solution of the problem (CP k (α)), obtained from (CP k ) when α is fixed. Let Then, we call the matrix X k = X k (α * ), the best solution of the problem (CP k ).The solution of the bilevel transportation problem (BCP) is that X k for which we have g(X k ) = min{g(X l ) : l ∈ {1, . . ., q}}.
In order to illustrate the above algorithm we conclude with the following numerical example.The set H is