Lagrange Multipliers Erasmus Mobility Program (24Apr2012) Pollack Mihály Engineering Faculty (PMMK) University of Pécs João Miranda ;

24-Apr-12Lagrange Multipliers2 Introduction: maximization of 2-variable function subject to one restriction; Generalization: maximization of n-variable function subject to m restrictions Tables & Chairs problem.

24-Apr-12Lagrange Multipliers3 Optima of multi-variable functions when submitted to several constraints. Problem: –Obtain the maxima/minima of the function, –But the variables x and y have to satisfy the equation:

24-Apr-12Lagrange Multipliers4 Assuming y(x) as implicit function of x, the derivative of the composite function u(x,y), (necessary condition) Or, Lagrange Multipliers

24-Apr-12Lagrange Multipliers5 The derivation of the implicit function, y(x): then, Lagrange Multipliers

24-Apr-12Lagrange Multipliers6 Conjugating the two zero equations: where is known as Lagrange multiplier ( ). ( Re-allocating terms, Lagrange Multipliers

24-Apr-12Lagrange Multipliers7 The multiplier is selected in conformity with, and it is also necessary that: ( Lagrange Multipliers

24-Apr-12Lagrange Multipliers8 Thus, in optimal points, the three following equations are simultaneously verified (necessary conditions to the existence of constrained optima) : Notice that they represent the three partial derivatives in (x, y, ) of the function: Verify it! Lagrange Multipliers

24-Apr-12Lagrange Multipliers9 Procedure: 1.Build the Lagrangean function, L(x,y, ) ; 2.Set to zero the related first order partial derivatives; 3.Obtain the values (x, y, ) that are satisfying the system of equations. Lagrange Multipliers

24-Apr-12Lagrange Multipliers10 Problem (general): –Obtain the maxima/minima of the function, –But the n variables have to simultaneously satisfy the set of m (m<n) equations: Lagrange Multipliers

24-Apr-12Lagrange Multipliers11 Procedure: 1.Build the Lagrangean function, ; Lagrange Multipliers

24-Apr-12Lagrange Multipliers12 (...) Procedure: 2.Set to zero the n first order partial derivatives in the n variables, and... Lagrange Multipliers

24-Apr-12Lagrange Multipliers13 (...) Procedure: Notice that the first order partial derivatives in order to the m multipliers is driving the m constraints, Lagrange Multipliers

24-Apr-12Lagrange Multipliers14 (...) Procedure: 3.Obtain the values set that simultaneously satisfy the group of (n + m) equations. That is, Lagrange Multipliers

24-Apr-12Lagrange Multipliers15 A furniture factory builds Tables ( t ) at a profit of 4 Euros per Table, and Chairs ( c ) at a profit of 3 Euros per Chair). Suppose that only 8 short ( s ) pieces and 6 large ( l ) pieces are available for building purposes, what combination of Tables and Chairs do you need to build to make the most profit?. If the availability of the short pieces is 8008 and the availability of the large pieces is 6007, how many Tables and Chairs do you need to build to make the most profit? Tables & Chairs (T&C)

24-Apr-12Lagrange Multipliers16 Note: modeling is based in the proportionality, aditivity and divisibility between the produced quantities of t and c and profit and utilization of l and s components.. Tables & Chairs (T&C)

24-Apr-12Lagrange Multipliers17 Note: modeling is based in the proportionality, aditivity and divisibility between the produced quantities of t and c and profit and utilization of l and s components. Tables & Chairs (T&C)

24-Apr-12Lagrange Multipliers18 T&C: Lagrange Multipliers Maximize the profit function, Luc(t,c), satisfying the conditions concerning the availability of the l and s components.. Problem: –Obtain the maximum value of profit function, –But the variables t and c shall satisfy the availability of l and s:

24-Apr-12Lagrange Multipliers19 Procedure: 1.Build the Lagrangean function, L(t, c, 1, 2 ) ; 2.Set to zero the 4 first order partial derivatives; 3.Obtain the values (t, c, 1, 2 ) that are satisfying the system of equations. T&C: Lagrange Multipliers

24-Apr-12Lagrange Multipliers20 Procedure: 1.Build the Lagrangean function, L(t, c, 1, 2 ) ; T&C: Lagrange Multipliers

24-Apr-12Lagrange Multipliers21 (...) Procedure: 2.Set to zero the 4 first order partial derivatives, and notice that those related to the multipliers ( 1, 2 ) are driving the original constraints: and... T&C: Lagrange Multipliers

24-Apr-12Lagrange Multipliers22 (...) Procedure: 3.Obtain the values (t, c, 1, 2 ) that are simultaneously satisfying the 4 equations: Re-allocating the terms of the system of equations, T&C: Lagrange Multipliers

24-Apr-12Lagrange Multipliers23 (...) Procedure: 3.Obtain the values (t, c, 1, 2 ) that are simultaneously satisfying the 4 equations: Applying the Cramer’s Rule to the 1. st subsystem, T&C: Lagrange Multipliers

24-Apr-12Lagrange Multipliers24 (...) Procedure: 3.Obtain the values (t, c, 1, 2 ) that are simultaneously satisfying the 4 equations: Applying the Cramer’s Rule to the 2. nd subsystem, T&C: Lagrange Multipliers

24-Apr-12Lagrange Multipliers25 (...) Procedure: 3.Obtain the values (t, c, 1, 2 ) that are simultaneously satisfying the 4 equations : Then, Optimal Solution! T&C: Lagrange Multipliers Edit LINDO: max 4t + 3c subject to l1) 2t + 1c <= 6007 l2) 2t + 2c <= 8008 END

24-Apr-12Lagrange Multipliers26 Lagrange Multipliers (synthesis) Introduction: maximization of 2-variable function subject to one restriction; Generalization: maximization of n-variable function subject to m restrictions Tables & Chairs problem.