On Boundness Conditions for the Set of Feasible Points of Systems of Linear Inequalities

Abstract In a linear programming problem involving maximization (or minimization) of the objective function the set of feasible points is often required to be bounded above (or below). A criterion based on the simplex method which requires the constraints coefficients of the entering variable to be zero or negative for the set of feasible points to be unbounded is often used. In this paper, the necessary and sufficient conditions for the set of feasible points of the system of linear inequalities to be bounded are stated and proved. These conditions which do not require the knowledge of the entering variable are illustrated with examples.


Introduction
The Boundedness of a set of feasible points of a system of linear inequalities is paramount in the solution of linear programming problems.The operations researchers always aim at developing models that would have finite optimal solutions.Unfortunately, some linear programming problems often results in an unbounded optimal feasible solution which occurs due to two major reasons, namely, when the set of feasible points is unbounded, and when the gradient of the ray directing into the cone giving the rays is orthogonal to the diverging hyperplane Eiselt, et al. (1987).
The existence of the unbounded optimal solutions in practical problems always indicates that a mistake has been made in modeling.It would therefore be unwise to formulate problems that has unbounded optimal solutions.For a finite optimal solution to exists, the set of feasible points is often required to be bounded above in the case of a maximization problem or bounded below in the case of a minimization problem.
The bounded set of feasible points is not always apparent, especially, when it is not possible to display the set graphically.Thus it is important for the Operations Researchers to know beforehand the conditions for the problems to be bounded.A criterion based on the simplex method which requires the constraints coefficients of the entering variable to be zero or negative for the set of feasible points to be unbounded is often used.Our interest in this paper is to derive the necessary and sufficient conditions for the system of linear inequalities to have a bounded set of feasible points.

Preliminaries
Consider the system of m linear inequalities in n variables Let (2) be the halfspace for each constraint i.Then the set, may be empty or nonempty.If F r is nonempty it defines the set of feasible points.
The set F r is said to be bounded if there exists a finite number c ∈ R so that (Fikhtengol's, 1979).

Conditions for the Set of Feasible Points to be Bounded
Consider the following linear programming problem The following lemma provides the simplex criterion for the objective function of linear programming problem (5) to be unbounded.
Lemma 1 (Shapiro, 1979) If, for a basic feasible system, there is a nonbasic variable x s with the properties cs < 0 and āis ≤ 0, for i = 1, 2, • • • , m, then the objective function of the linear programming problem can be driven to −∞.
Theorem 3.1 (Effanga, 2009) Consider the constraint matrix If there exists a column j * , 1 ≤ j * ≤ n, such that a i j * ≤ 0, for all 1 ≤ i ≤ m, then the set of feasible points F r is unbounded.
Writing the above constraints as and adding a positive number λ to x j * , we obtain Using a P-norm defined as follows This implies that So F r is unbounded.
Theorem 1 is equivalent to the unboundedness criterion for the simplex algorithm in Shapiro (1979), Hillier and Lieberman (2005), Eieselt et al. (1987) or Taha (2005).This condition is only necessary but not sufficient for the detection of the unboundedness of the set of feasible points.
Proof: If max 1≤i≤m {a i j } ≤ 0, for any j * , then a i j * ≤ 0, for all 1 ≤ i ≤ m.This implies that F r is unbounded by Theorem 1.
Hence, if F r is bounded, then, max 1≤i≤m {a i j } > 0, for all 1 ≤ j ≤ n.
Example 1 Consider the set of feasibility points defined by the simultaneous linear inequalities The above set of feasibility points is unbounded since a 11 < 0 and a 21 < 0 (See figure 1) <Figure 1> The following example illustrates the fact that theorem 1 is not a sufficient condition for the set of feasible points to be unbounded.
Example 2 Consider the set of feasibility points defined by the simultaneous linear inequalities The set of feasible points represented by the above system of linear inequalities is unbounded, but there is no column of the constraints matrix with negative entries throughout (see figure 2).

<Figure 2>
Example 3 Consider the set of feasible points represented by the system of linear inequalities below, The above set of feasible points is bounded (see figure3), hence, <Figure 3> Examples 2 and 3 illustrate the fact that corollary 1 is not a sufficient condition for the set of feasible points to be bounded.We now present a necessary and sufficient condition for the set of feasible points to be bounded.
Theorem 3.2 (Effanga, 2009) The constraints, Ax ≤ b, determines a bounded set of feasible points, if and only if there exists a c ∈ R n , c > 0 and Proof:Let Ax ≤ b be bounded, then there exists T ∈ R + so that Conversely, we now show that if Ax ≤ b ⇒ cx ≤ T , ∀x ∈ F r , then F r is bounded.
Without loss of generality, we assume that there exists a row A i * of the constraints matrix A such that c j = a i * j > 0, 1 ≤ j ≤ n and 1 ≤ i * ≤ m and b i * = T .
From the constraints Ax ≤ b, we get Theorem 2 provides a necessary and sufficient condition for the set of feasible points of linear system of inequalities to be bounded.
Hence, the set of feasible points is bounded (see figure 4).

<Figure 4>
Example 5 Consider the set of feasibility points represented by the simultaneous linear inequalities below, On multiplying (III) by -1 yields Multiplying (I) by 1.5 and adding to (II) yields 0.5x 1 + 0.5x 2 ≤ 9 (V) Adding (II) and (IV) we have Multiplying (V) by 5 and adding to (VI) we have Hence, the set of feasible points is bounded (see shaded region in figure 5).

Conclusion
The unboundedness condition given in Theorem 1 is equivalent to the simplex criterion for the unboundedness of a set of feasible points of the system of linear inequalities.In simplex algorithm the entering variable need to be known before the unboundedness can be detected, and in most cases this will happen after a number of iterations have been performed.In our case no knowledge of the entering variable is required to determine whether the set of feasible points is unbounded or not.A necessary and sufficient condition for the system of linear inequalities to be bounded given in Theorem 2 can only be applied when system of linear inequalities are stated with a less than or equal to sign.Fortunately, any system of linear inequalities can be put in this form, thus making the condition valid for any system of linear inequalities.

Figure 1 .
Figure 1.Showing the unbounded set of feasible points (shaded)