Math Lab
← All labs

1-D LP: Max p·x s.t. x ∈ [0, M]

Linear Programming · Class XII

A single-variable LP. The optimum sits at an endpoint of the feasible interval , see which one.

2
type a value
-55
10
type a value
120
Live values
  • Max value20
  • Optimal x*10
  • Min value0
xy
  • Objective y = p x

Formulas in this lab

  • Max value
    max⁡(0,pM)\max(0, p M)
  • Optimal x*
    p≥0⇒x∗=M; else x∗=0p \ge 0 \Rightarrow x^* = M;\ \text{else } x^* = 0
  • Min value
    min⁡(0,pM)\min(0, p M)
Tip: If p < 0 the max is at x = 0 (corner); if p > 0 the max is at x = M.

Frequently asked questions

▶What is the simplest linear programming problem?

Maximize $z = px$ subject to $0 \le x \le M$. If $p > 0$, optimum is at $x = M$ with value $pM$. If $p < 0$, optimum is at $x = 0$ with value $0$. The maximum always sits at a corner of the feasible interval.

▶How do I use the 1-D LP lab?

Slide the objective slope $p$ and the upper bound $M$. The lab returns the max value $\max(0, pM)$ and the optimal $x^*$. Try $p = 2$, $M = 10$: max is $20$ at $x^* = 10$. Flip $p$ to $-2$ and the optimum jumps to $x^* = 0$.

▶Why is the optimum always at a corner in LP?

Linear objectives on a convex feasible region (a polygon) achieve max and min at vertices. This is the corner-point theorem and is the basis of the simplex method. In 1-D the corners are just the two endpoints, making the principle crystal-clear.

▶Where is LP used in real life?

Diet planning: minimize cost subject to vitamin requirements. Production planning: maximize profit subject to machine hours. Indian Railways uses LP for train scheduling and crew assignment. Practising 1-D LP builds intuition for the higher-dimensional cases on JEE.