Answer to Question #237086 in Operations Research for lavanya

Question #237086

. A tailor has the following material available: 16 sq. yd. cotton, 11 sq. yd. silk and 15 sq. yd. wool. A suit requires 2 sq. yd. cotton, 1 sq. yd. silk and 1 sq. yd. wool. A gown requires 1 sq. yd. cotton, 2 sq. td. Silk and 3 sq. yd. wool. If a suit sells for Rs. 300/- and a gown for Rs. 500/-, how many of each garment should the tailor mak to obtain the maximum amount of profit?


1
Expert's answer
2021-09-20T04:52:14-0400

Let x=x= the number of suits, y=y= the number of gowns.

Maximize Z=300x+500yZ=300x+500y

subject to


2x+y162x+y\leq16

x+2y11x+2y\leq11

x+3y15x+3y\leq 15

x0,y0x\geq0, y\geq0

OA:x=0,0y5,Z=500yOA: x=0, 0\leq y\leq5, Z=500y

AB:y=513x,0x3,Z=4003x+2500AB: y=5-\dfrac{1}{3}x, 0\leq x\leq3, Z=\dfrac{400}{3}x+2500

BC:y=11212x,3x7,Z=50x+2750BC: y=\dfrac{11}{2}-\dfrac{1}{2}x, 3\leq x\leq7, Z=50x+2750

CD:y=162x,7x8,Z=700x+8000CD: y=16-2x, 7\leq x\leq8, Z=-700x+8000


OD:y=0,0x8,Z=300xOD: y=0, 0\leq x\leq8, Z=300x


ExtremeLines throughObjective functionpointexreme pointvalueO(0,0)x=0,Z=500(0)y=0=0A(0,5)x=0,Z=500(5)y=513x=2500B(3,4)y=5x/3,Z=50(3)+2750y=11/2x/2=2900C(7,2)y=11/2x/2,Z=700(7)+8000y=82x=3100D(8,0)y=82xZ=300(8)y=0=2400\def\arraystretch{1.5} \begin{array}{c:c:c} Extreme & Lines\ through & Objective \ function \\ point & exreme\ point & value\\ \hline O (0,0) & x=0, & Z=500(0) \\ & y=0 &=0\\ \hdashline A (0,5) & x=0, & Z=500(5)\\ & y=5-\dfrac{1}{3}x & =2500\\ \hdashline B (3,4) & y=5-x/3, & Z=50(3)+2750\\ & y=11/2-x/2 & =2900\\ \hdashline C (7,2) & y=11/2-x/2, & Z=-700(7)+8000\\ & y=8-2x & =3100\\ \hdashline D (8,0) & y=8-2x & Z=300(8)\\ & y=0 & =2400\\ \hdashline \end{array}

The maximum value of the objective function Z=3100Z=3100  occurs at the extreme point (7,2).(7, 2).

Hence, the optimal solution to the given LP problem is :

 x=7,y=2x=7,y=2  and maxZ=3100.max Z=3100.



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