Transform the following Linear Programming Problem into the standard form where all constraints are of equality type:
Maximize z= 2x1+x2-5x3+3x4
Subject to the constraints:
3x1+2x2 ≤ 15
4x1+5x2 ≥ 20
X1+x2-x3+2x4=10
2 ≤ 2x1+4x2-x3 ≤ 30
Xi ≥ 0; i=1 to 3 x4 unrestricted in sign.
Answers
Answer:
Linear programmes can be written under the standard form:
Maximize ∑n
j=1 c jx j
Subject to: ∑n
j=1 ai jx j ≤ bi for all 1 ≤ i ≤ m
x j ≥ 0 for all 1 ≤ j ≤ n.
(9.1)
All constraints are inequalities (and not equations) and all variables are non-negative. The
variables x j are referred to as decision variables. The function that has to be maximized is
called the problem objective function.
Observe that a constraint of the form ∑n
j=1 ai jx j ≥ bi may be rewritten as ∑n
j=1(−ai j)x j ≤
−bi. Similarly, a minimization problem may be transformed into a maximization problem:
minimizing ∑n
j=1 c jx j is equivalent to maximizing ∑n
j=1(−c j)x j. Hence, every maximization
or minimization problem subject to linear constraints can be reformulated in the standard form
(See Exercices 9.1 and 9.2.).
A n-tuple (x1,...,xn) satisfying the constraints of a linear programme is a feasible solution
of this problem. A solution that maximizes the objective function of the problem is called an
optimal solution. Beware that a linear programme does not necessarily admits a unique optimal
solution. Some problems have several optimal solutions while others have none. The later case
may occur for two opposite reasons: either there exist no feasible solutions, or, in a sense, there
are too many. The first case is illustrated by the following problem.
Maximize 3x1 − x2
Subject to: x1 + x2 ≤ 2
−2x1 − 2x2 ≤ −10
x1,x2 ≥ 0
(9.2)
which has no feasible solution (See Exercise 9.3). Problems of this kind are referred to as
unfeasible. At the opposite, the problem
Maximize x1 − x2
Subject to: −2x1 + x2 ≤ −1
−x1 − 2x2 ≤ −2
x1,x2 ≥ 0
(9.3)
has feasible solutions. But none of them is optimal (See Exercise 9.3). As a matter of fact, for
every number M, there exists a feasible solution x1,x2 such that x1 − x2 > M. The problems
verifying this property are referred to as unbounded. Every linear programme satisfies exactly
one the following assertions: either it admits an optimal solution, or it is unfeasible, or it is
unbounded.
Geometric interpretation.
The set of points in IRn at which any single constraint holds with equality is a hyperplane in
IRn. Thus each constraint is satisfied by the points of a closed half-space of IRn, and the set of
feasible solutions is the intersection of all these half-spaces, a convex polyhedron P.
Because the objective function is linear, its level sets are hyperplanes. Thus, if the maximum
value of cx over P is z∗, the hyperplane cx = z∗ is a supporting hyperplane of P. Hence cx = z∗
contains an extreme point (a corner) of P. It follows that the objective function attains its
maximum at one of the extreme points of P.
Step-by-step explanation:
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