Constrained Optimization And Lagrange Multipliers

5.12, (I) Use Lagrange multipliers to optimize the following functions subject to the given constraint, and (2) estimate the effect on the value of the objective function from a 1-unit change io the constant of the constraint a) z m 4x* - 2xy + 6y* subject to x + y » 72

1) Set the constraint equal to zero, multiply it by A, and add it to the objective function, to obtain

Z-4*z-2xy + 6yi+ A(72-x-y) The first-order conditions arc

10* - 14y ■ 0 x«1.4y Substitute x = I Ay in (5.2V) and rearrange

1 Ay+y**72 y0m 30 Substitute y0 = 30 m the previous equations to find that at the critical point

55 42 y„ - 30 Ao = 276 Thus. Z - 4(42^ - 2(42X30) + 6<30;.a + 276(72 - 42 - 30) - 9936.

2) With A = 276. a 1-unit increase in the constant of the constraint will lead to an increase of approximately 276 in the value of the objective function, and Z — 10.212.

b) f(x.y) - - a*3 + 5xy - 6/ + 12y subject to 3x +y = 170

1) The Lagrangian function is

Fk - 170 - 3x - y = 0 [5.32) Multiply (5 31) by 3 and subtract from (5.30) to eliminate A.

-21*+ 41/-10-0 {5.33) Multiply (5J2) by 7 and subtract irom (5J3) to eliminate x.

Then substituting y9 - 25 into the previous equations shows that at the critical point

2) With A - -46$, a 1-unit increase in the constant of the constraint will lead to a decrease of approximate!) 46$ in the value of the objective fuaction. and F" -3206.33.

Chen subsututing = 14 in the previous equations gives

■t, - 14 y0 - 14 Zo - 28 Afl - 43.904 Fa - 614.656

2) F, - F0 + A, - 614.656 + 43.904 - 658.560 Sec Problem 12.28 for the second-order conditions

<0 f(x.y.i) = 5*y + &rz + 3yz subject to 2xyz - 1920

2yr z y

2xz z x

2xy y x

Equate A*» in (5.42) and (5.43) to eliminate 2.51 z.

4 15 * . < iJS _ e — -Vr = 1.5v x = —v y x 4

rhen substitute x - (1.5/4)y and r - (2.5/4)y in (5.41).

and the critical values are x0 «6. y„ ■ 16. = 10. and A0 = 0.5.

5.13. In Problem 5.12(a) it was estimated that if the constant of the constraint were increased by 1 unit, the constrained optimum would increase by approximately 276. from 9936 to 10,212. Check the accuracy of the estimate by optimizing the original function j = 4jr - 2xy + 6y* subject to a new constraint x + y = 73.

Z ■ Ax2 - 2xy + 6y* + A(73 -x - y) 2. = 8r - 2y - A » 0 Z, = -2x + I2y - A - 0 Z, = 73 - x - y = 0

Simultaneous solution gives - 42.58. y0 - 30.42, Ao - 279.8. Thuv Zq - 10,213.9, compared to ihe 10,212 estimate from the original A. a difference of 1.9 units of 0.02 percent.

5.14. Constraints can also be used simply to ensure that the two independent variables will always be in constant proportion, for example, x = 3y. In this case measuring the effect of A has no economic significance since a 1-unit increase in the constant of the constraint would alter the constant proportion between the independent variables With this in mind, optimize the following functions subject to the constant proportion constraint:

Z « 4x* - 3Lr + 5xy -8y + 2y' + A(x - 2y) 7. - R» - 3 + 5y + A - 0 Z, = 5x - 8 + 4v - 2A = 0 Z» «= x — 2y ■ 0

When solved simultancouily. • 0.5, y0 •» 0.25. and A© - -2.25. Thus, Zo » -1.75.

b) : = -Sx2 + 7i + 10*y + 9> - 2/ subject to y = 5c

The Ugrangian function is Z » - Sar2 - 7jt + IOrv + 9y - 2/ + A(5* - y)

Z, - -10*+ 74 lOy + 5A ■ 0 Z, - 10* + 9 - 4v - A - 0 2» ■ 5* - y • 0

Solving simultaneously. *<» = 5.2, y„ = 26. and A<, = -43. Thus. Z,, = 135.2. Sec also Probiims 12.19 to 1228.

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