A Bilevel Programming Problem (BPP) is a decision problem where the vector variables and are controlled by two decision-makers: the leader and the follower. Variables (resp. ) are variables of decision at the upper (resp. lower) level. This structure of hierarchical optimization appears in many applications when the strategic of the lower level depends on the strategic of the upper level.
Mathematically, solving a BPP consists of finding a solution of the problem at the upper level called the leader’s (or outer’s) problem;
where for each value of , is the solution of the problem at the lower level, which is called the follower’s (or inner’s) problem;
with , ; are the objective functions of the upper (resp. lower) level; are the constraint functions of the upper (resp. lower) level.
In the literature, the BPP and the problem with multiple objectives at the upper level or at the lower level are presented as a class of bilevel problems and are at the center of research of some authors such as   . In the quasiconcave case, Herminia et al.  include an optimization problem at the upper (resp lower) level, in which the objective functions are quasiconcave and linear. Fatehem et al.  present Particle Swarm Optimization (PSO) algorithm for solving the bilevel programming problem with multiple linear objectives at the lower level while supposing the objective function at the upper level quasiconcave. They conclude that the feasible region of the problem consists of faces of the polyhedron defined by the constraints. O. Pieume, L. P. Fotso et al.   study Bilevel Multiobjective Programming Problem (BMPP). For the linear case, they establish equivalence between the feasible set of a bilevel multiobjective linear programming and the set of efficient points of an artificial multiobjective linear programming problem. The same authors   show how to construct two artificial multiobjective programming problems such that any point that is efficient for both problems is an efficient solution of a BMPP. Pu-Yan Nie  studies bilevel programming problem where the problem at the lower level is a multiobjective programming problem by using weighting methods to analyze the constraint conditions for multiobjective programming problem. Farahi et al.  extend the kth-best methods to solve multiobjective linear bilevel programming problems by using fuzzy set theory and fuzzy programming to convert the multi-objective linear bilevel programming (MOLBLP) problem to a linear bilevel programming problem.
Clearly, there are very few approaches in the literature that deal with bilevel multiobjective problems. According to Pieume et al.  , it is not easy to find efficient solution of BMPP. In  , the authors propose to approximate the efficient set of multiobjective programming problem by the weakly efficient set and give an approach to generate a representative subset of efficient set by using well known schemes   .
In this paper, we are interested in finding the solution of a quasiconcave bilevel programming problem (QCBPP). After the formulation of a bilevel multiobjective programming problem (BMPP), we characterize its leader objective and its feasible set. Then, we show some necessary and sufficient conditions to establish that a convex union of set of efficient point is an efficient set of the QCBPP. Based on this result a QCBPP is formulated and solved. A numerical example is provided to illustrate our approach.
This paper is organized as follows: in the next section, we present some concepts and results in multiobjective programming. In section 3, we define and formulate a BMPP. We give in section 4, a characterization of QCBPP. In section 5, we illustrate our approach with a numerical example. Section 6 concludes the paper.
2. Multiobjective Programming Problem
Here, we give some concepts and results of multiobjective programming that will be used throughout the paper.
Preliminaries and Notations
A multi-objective programming problem is formulated in general as follows:
with where the are the objective functions for all and is the feasible set. In order to solve (MOPP), it is necessary to define how objective function vectors should be compared for different alternatives . We must define on h(U) the order that should be used for this comparison. Due to the fact that, for there is no canonical (total) order in . Calice Pieume and al  propose to define partial orders on .
Let be an arbitrary cone. They show that the binary relation defined in C by: , achieves a partial order introduced by closed pointed convex cones that are the most used.
Consider the linear optimization problem (LOP)
where is the weight of the i-th objective and defines the importance of each objective.
Geoffrion  shows that for fixed for all i, if is an optimal solution of the LOP then is efficient solution of MOPP. Greffrion  also shows that if U is a convex set and a concave function on U for all i then is a weakly efficient solution if and only if is an optimal solution of the LOP for a having positive components. If is Pareto-optimal then is called Non-dominated point.
Throughout the rest of the paper, the set of efficient points of a multi-objective optimization problem defined by a vector value function h on a feasible set U with respect to a cone C will be denoted: .
3. Definition of the Problem and Formulation of BMPP
Consider the problem (1) called the leader’s problem formulated as follows:
where for each value of , is the solution of the problem (2) called the follower’s problem;
, (resp. ) are the decision variable vectors controlled by the leader (resp. the follower). . and are the objective functions of the leader’s problem and follower’s problem respectively; and are the constraint functions of the leader’s problem and follower’s problem respectively.
Let us consider a bilevel programming problem (BPP) that comprises at the upper level the leader’s problem (1) and at the lower level the follower’s problem (2). The feasible region of the BPP of the first level is implicitly determined by the follower’s problem (2). This bilevel programming problem is called bilevel multiobjective programming problem (BMPP) and is defined as follows:
Let denote the feasible region of the problem (2). The solution set of the follower’s problem denoted by:
is called lower-level reaction set for each decision x of the upper level and is defined as the set of Pareto-optimal points.
Let define a lower level solution y for every feasible x such that:
is a parameter of the follower’s problem (2).
Let consider and with be compact set. The bilevel multi-objective programming problem (BMPP) can be reformulated as follows:
Let denote by the feasible space (also called induced set) of BMPP given by:
The optimistic formulation of BMPP is given by:
For a fixed , if is a Pareto optimal solution of the follower’s problem, then is a feasible solution to the BMPP.
4. Characterization of QCBPP
Let F be the objective function of the BMPP.
Definition 1. The objective function F of the BMPP defined on a convex subset of with values in R is quasiconcave if for all real k the whole
Lemma 1. Let be a convex subset of of interior non empty and be quasiconcave then , ,
Let’s suppose that F is quasiconcave on the convex , and . Let’s apply the definition of the quasiconcavity of F to . One has , that is to say , which is convex by hypothesis on F. Therefore, : In other words, or and therefore
The lemma 1 establishes that components of F are quasiconcave functions on the convex set .
Theorem 1. Let be a nonempty convex and compact subset of and let be any function.
If F is quasiconcave and continuous, then there exists an extreme point of the polyhedron which is an optimal solution of the BMPP.
Let suppose F quasiconcave and continuous and show that is optimal solution of the BMPP.
Consider a non-empty compact set of optimal points and Let denote the optimal solution of the BMPP.
Let . Since F is a quasiconcave function, there exists such that for all , . By hypothesis, F is continuous on implies that . For with , , is an optimal solution of the BMPP.
Definition 2. The feasible point is the optimal solution of the BMPP if for each point .
For BMPP, it is noted that a solution is optimal for the upper level problem if and only if is an optimal solution for the lower level problem with .
4.1. Necessary and Sufficient Conditions
Given a fixed value of , the problem (2) can be rewritten as follows:
In the following, let be a cone. The feasible region of the follower’s problem is the area . is the efficient set.
Theorem 2. Let with and be non-empty efficient subset of . The following result holds.
Let . Then there exists such that
, thus implies that
Theorem 2 permits to say that is the efficient set of the
Lemma 2. If F verifies
for all and then
Let suppose F quasiconcave and show that is convex.
Let and .
for all real k. Thus, . Therefore , implies
Theorem 3. Let with and be non-empty efficient subset of . The following result holds.
implies that implies that
4.2. Formulation of QCBPP
Let and consider the following constructed follower’s problem:
Let , The result of Theorem 4 holds by Theorem 2 and Theorem 3.
Since the set is convex, solving the BMPP is then equiva-
lent to solving the quasiconcave problem:
Definition 3 If is a feasible solution to the QCBPP and there are no
such that , then is a
Pareto optimal (efficient) solution to the QCBPP, where the binary relation
defines a partial order in .
Theorem 5. is an optimal solution of BMPP if only if
is an efficient solution to the QCBPP.
Þ) Let suppose optimal solution of BMPP and show that is efficient solution of QCBPP.
Let . Since F is a continuous function on, ,
. According to theorem 3 and
by definition 3 there is no such that
. Then . Hence
is efficient solution of QCBPP.
(Ü Let suppose is efficient solution of QCBPP. Let us show that is optimal solution of the BMPP.
Let is efficient solution of QCBPP with
such that: 1) . Taking into ac-
count the theorem 4, one has : 2) for all .
Due to the relations 1) and 2), . Therefore is optimal solution of BMPP.
4.3. The Efficient Subset of QCBPP
Let , the efficient subset of ,
such that for a fixed , is a minimizing solution to the problem
Let an optimal solution of the following problem:
If is fixed and for all , is an optimal solution of (4) then is efficient solution of (3).
If is fixed and for all , is an optimal solution of (4) then is weakly efficient solution of (3).
That is, is an efficient solution as well as weakly efficient solution in
Therefore, represents the efficient subset in which is an efficient solution to the QCBPP.
5. Illustrative Example
This example is taken from  . Let consider the leader’s problem:
and the follower’s problem:
, (resp. ) are the decision variable vectors controlled by the leader (resp. the follower).
The two multi-objective problems used are:
Let be the constraint region of the lower level problem
and be the solution set of the lower-level problem.
Consider the Pareto optimal solution of the follower’s problem and with a compact set, the BLMPP becomes:
This problem can be formulated as:
, the feasible space (also called induced set) of the reformulated BMPP is:
Lemma 1 establishes that F is quasiconcave function on the convex set .
According to the theorem 1, is an optimal solution to the QCBPP for fixed optimal solution of the follower’s problem.
Therefore, the follower’s problem is as follows:
The feasible region of the follower’s problem is the area
Here the cone , hence .
With r = Sup W = 7 (Superior of W = 7), one has are non-empty efficient subsets of the follower’s problem
The objective function of the BMPP very-
fies for all
and F is quasicon-
cave function on convex.
with we have
. implies that
We therefore have according to the Theorem 2:
The follower’s problem is constructed as follows:
The set is convex and solving the BMPP is equivalent to
solving the quasiconcave problem
Theorem 5 says that (1.5;,1.5, 0) is an optimal solution of the BMPP if and only if it is an efficient solution to the QCBPP.
implies that and is a maximizing solution to the problem:
is an optimal solution of the following problem:
For with , is an optimal solution of (II) and is efficient solution of (I). Also, with is an optimal solution of (II) and is weakly efficient solution of (I). Thus, (1.5,1.5, 0) is an efficient solution as well as weakly efficient solution in and therefore represents the efficient subset in which is the solution to the QCBPP.
In this paper, we have uniquely defined a lower level solution for every upper level feasible solution as a parameter of the follower’s problem. We have formulated a Bilevel Multiple Programming Problem (BMPP), of which we have considered the quasiconcave objective function and showed that there was an extreme point of the feasible space that was an optimal solution of the BMPP. We have proven a theorem, suggesting that the optimal solution of the BMPP is an efficient solution to the QCBPP. Based on this result, we presented an efficient solution which was a weakly efficient solution in the efficient subset as well. We proved that this efficient solution was the solution of the QCBPP. Thus, we concluded that solving BMPP was equivalent to solving the QCBPP.