Allowable Ranges

Allowable RHS Range

The same basis can survive a range of RHS changes. This lesson derives the first resource interval by keeping the exact basis solution nonnegative.

highlighted = computed this step

Basis direction

Changing the first RHS moves x by 2/3 per unit and y by -1/3 per unit. Motivation: the inverse basis translates RHS changes into primal movement.

Δx=2/3Δb1Δy=−1/3Δb1\Delta x=2/3\Delta b_1\quad \Delta y=-1/3\Delta b_1
First RHS allowable intervalThe interval is derived from keeping the same basis point nonnegative.first RHS sensitivitydxdylowhighdirection2/3-1/3-24endpoints0240

Allowable interval

The first RHS delta can range from -2 through 4. Why: within that interval, the same basis point remains nonnegative.

Δb1∈[−2,4]\Delta b_1\in[-2,4]
First RHS allowable intervalThe interval is derived from keeping the same basis point nonnegative.first RHS sensitivitydxdylowhighdirection2/3-1/3-24endpoints0240

Endpoint points

At the low endpoint the point is (0, 2); at the high endpoint the point is (4, 0). Interpretation: the range ends exactly when one decision variable reaches its bound.

endpoints 2,4\text{endpoints }2,4
First RHS allowable intervalThe interval is derived from keeping the same basis point nonnegative.first RHS sensitivitydxdylowhighdirection2/3-1/3-24endpoints0240

Certificate note

RHS sensitivity is local: the shadow-price prediction is exact while the same basis remains primal feasible and dual feasible. Outside that range, the LP must be recomputed. Pixel positions are rounded for layout; every number shown is exact.

allowable range is a feasibility range for the same basis\text{allowable range is a feasibility range for the same basis}
First RHS allowable intervalThe interval is derived from keeping the same basis point nonnegative.first RHS sensitivitydxdylowhighdirection2/3-1/3-24endpoints0240