Answer to Question #283646 in Operations Research for Eden

Question

Bedele Brewery produces Ale and Beer. Suppose that the productions are limited by scarce resources of Corn, Hops and Barely malt. To make Ale, 5 kg of Corn, 4 Kg of Hops and 35 Kg of Barely malt are required. To make Beer, 15 Kg of Corn, 4 Km of Hops and 20 Kg of Barely malt are required. Suppose that only 480 Kg of Corn, 160 Kg of Hops and 1190 Kg of Barely malt are available. The Brewery plans to enjoy a Profit of Birr 13 for each Kg of Ale and Birr 23 for each Kg of Beer.

Required:

Formulate the linear programming model

How many Ale and Beer should the Brewery Company produce in order to maximize the returns? Using the graphical method. 

What is the total profit? 

Is there any slack?

Answer 1

Let x₁ be the amount of Ale to be produced.

And x₂ be the amount of Beer to be produced.

Profit function: Maximize $Z=13x_1+23x_2$

Subject to:

$5x_1+15x_2\le 480 \\4x_1+4x_2\le 160 \\35x_1+20x_2\le 1190 \\x_1\ge0,x_2\ge0$

Plotting and graphing, we get,

OABCD is the feasible region.

$O(0,0): Z=13(0)+23(0)=0 \\A(0,32):Z=13(0)+23(32)=736 \\B(12,28):Z=13(12)+23(28)=800\ (maximum) \\C(26,14):Z=13(26)+23(14)=660 \\D(34,0):Z=13(34)+23(0)=442$

Thus, the maximum profit is 800 at $x_1=12\ kg,x_2=28\ kg$

The Slack or Surplus column tells us how close we are to satisfying a constraint as an equality. This quantity, on less-than-or-equal-to (≤) constraints, is generally referred to as slack. On greater-than-or-equal-to (≥) constraints, this quantity is called a surplus.

In this problem, we have less-than-or-equal-to (≤) constraints, so yes, there is slack