Find graphical sensitivity analysis to identify the range to make Linear Programming problem

Question

Refer the following LP formulation with unknown nos S: Max A + B s.t S(A) + B smaller or = 1 A,B bigger or = 0 Employ graphical sensitivity analysis to identify the range of S to make the LP problem: a) having an optimal solution b) being infeasible c) bring inbound

Answer 1

(1) Assume S(A)=S multiplied by A.

Graphical representation is right triangle OPQ, where O(0,0), P(0,1), Q(1/S,0), so OPQ is the feasibility region.

(a) Optimal solution:

A, B≥0⇒S≥0⇒

              0<S<1⇒max(A+B)=1/S (A=1/S, B=0);

              S≥1⇒max(A+B)=1 (A=0, B=1);

              S=0⇒max(A+B)=1 (A any value, B=1).

(b) Infeasible: S<0.

(2) Assume S is a function of A.

(a) Optimal solution:

Range of S must include S(A)≥0 for some A≥0:

              0<S(A)<1⇒max(A+B)=1/S(A) (B=0, for some A);

              S(A)≥1⇒max(A+B)=1 (B=1, for some A);

              S(A)=0⇒max(A+B)=1 (B=1, for some A).

(b) Infeasible: S(A)<0 for all A.

EXAMPLE: S(A)=3A-A²-2, but this would be non-linear.

1≤A≤2⇒min(S(A))=0, max(S(A))=¼; A<1 or A>2⇒S(A)<0.

(a) max(A+B)=4 at A=3/2, B=0

(b) Infeasible when A<1 or A>2.

Still investigating for (c).