Q.A chemical company produces a chemical containing three basic elements A, B, C so that it has at least 16 liters of A, 24 liters of B and 18 liters of C. This chemical is made by mixing two compounds I and II. Each unit of compound I has 4 liters of A, 12 liters of B, 2 liters of C. Each unit of compound II has 2 liters of A, 2 liters of B and 6 liters of C. The cost per unit of compound I is Rs.800/- and that of compound II is Rs.640/-. Formulate the problem as L.P.P. and solve it to minimize the cost.

The chapter proves (without proof, as a stated theorem) that when the objective function of an L.P.P. attains its optimum (maximum or minimum) value over a convex-polygon feasible region, that optimum is always attained at one of the polygon's vertices (corner points) — this is the Corner-Point or Convex-Polygon Theorem, and it is what makes the graphical method possible: instead of checking every point of an infinite region, only the (finitely many) vertices need to be checked. The Corner-Point Method proceeds in five steps: (1) convert every constraint's inequation into its boundary equation and draw the line; (2) shade the half-plane each constraint demands and locate the region common to all of them — the feasible region; (3) find every vertex of this feasible region, either by reading intercepts directly off the graph or, more reliably, by solving each relevant pair of boundary-line equations simultaneously; (4) evaluate the objective function $z$ at every vertex f …