Answer to Question #197270 in Operations Research for Kurt

Question

Southern Sporting Goods Company makes basketballs and footballs. Each product is produced from two resources—rubber and leather. The resource requirements for each product and the total resources available are as follows:

Each basketball produced results in a profit of $12, and each football earns $16 in profit.

Formulate a linear programming model to determine the number of basketballs and footballs to produce in order to maximize profit.

Transform this model into standard form.

Solve the model formulated for Southern Sporting Goods Company graphically.

Identify the amount of unused resources (i.e., slack) at each of the graphical extreme points.

Answer 1

The resource requirements:

maximize: $Z=12x+16y$

subject to:

$3x+2y\le500$

$4x+5y\le800$

$x,y\ge0$

x is basketball product,

y is football product

Consider points: $A(0,160),B(128.5,57.2),C(157,0)$

For point A:

$Z=16\cdot160=\$\ 2560$ (slack: $x=500,y=640$ )

For point B:

$Z=12\cdot128.5+16\cdot57.2=\$\ 2457.20$ (slack: $x=371.5,y=742.8$ )

For point C:

$Z=12\cdot157=\$\ 2004$ (slack: $x=343,y=800$ )

So $Z_{max}=\$\ 2560$