Input-output model and linear programming

The input-output model, built by Wassily Leontief, traces how industries buy from each other: producing one unit of steel needs iron, coal and electricity, which themselves need inputs. If A is the matrix of input coefficients (input i needed per unit of output j), then gross output x must satisfy x = Ax + d, where d is final demand, so x = (I - A)^{-1} d. Linear programming solves a different problem: maximising or minimising a linear objective subject to linear constraints. Worked example: with A = [[0.2, 0.3], [0.4, 0.1]] and final demand d = (100, 50), the Leontief system gives gross outputs solving 0.8x1 - 0.3x2 = 100 and -0.4x1 + 0.9x2 = 50, which yields x1 = 175 and x2 = 133.33 (check: 0.8 x 175 - 0.3 x 133.33 = 140 - 40 = 100).