Chapter 3, Problem 14RP

Explanation of Solution

LP Formulation:

• Given data:

Method	Grade 6	Grade 8	Grade 10	Cost ($)
1 2	0.2 0.3	0.2 0.3	0.6 0.4	3.40 3.00
3	0.4	0.4	0.2	2.60

• Let $W_a,W_b,A_a$ and $A_b$ be the pounds of wheat and alfalfa used to produce feed sold at discount price.
• Let $W_1,W_2,A_1$ and $A_2$ be the pounds of wheat and alfalfa used to produce feed sold at non-discounted price.
• The objective is to maximize Feedco’s profit.
• Profit = (cost/lb) (pounds of wheat and alfalfa feed 1 sold at non-discounted price) + (cost/lb) (pounds of wheat and alfalfa feed 2 sold at non-discounted price) + (cost/lb) (pounds of wheat and alfalfa feed 1 sold at discounted price) + (cost/lb) (pounds of wheat and alfalfa feed 2 sold at discounted price) – (cost/lb) (total pounds of wheat purchased) – (cost/lb) (total pounds of alfalfa purchased)
• Profit= $1.5\left(W_1+A_1\right)+1.3(W_2+A_2)+1.25(W_a+A_a)+1(W_b+A_b)-0.5(W_1+W_2+W_a+W_b)-0.4(A_1+A_2+A_a+A_b)$
• Thus, the objective function is,
• Maximize
• z= $1.5\left(W_1+A_1\right)+1.3(W_2+A_2)+1.25(W_a+A_a)+1(W_b+A_b)-0.5(W_1+W_2+W_a+W_b)-0.4(A_1+A_2+A_a+A_b)$
• The following constraints are,
• Constraints 1: At most, 300lb of feed 1 can be sold.
• (Pounds of wheat produce feed 1) + (Pounds of alfalfa produce feed 1) ≤300
  • $W_a+A_a\le 300$
• Constraint 2: At most, 300lb of feed 1 can be sold.
• (Pounds of wheat produce feed 2) + (Pounds of alfalfa produce feed 2) ≤300
  • $W_b+A_b\le 300$
• Constraint 3: At most, 1000lb of wheat can be purchased.
• (Pounds of wheat used to produce feed 1) + (Pounds of wheat used to produce feed 2) ≤1000
  • $W_1+W_2+W_a+W_b\le 1000$
• Constraint 4: At most, 800lb of alfalfa can be purchased