92 (2)
Let be the number of grade 9 oranges O.J. uses for bags, be the number of grade 9 oranges O.J. uses for juice, be the number of grade 6 oranges O.J. uses for bags, and be the number of grade 6 oranges O.J. uses for juice. The objective is to maximize the profit. For juice, the profit will be ($1.50$1.05)*( + ), and for bags, the profit will be ($0.50-$0.20)*( + ). Thus, the total profit will be . The objective function is: Maximize The constraints are: Constraint 1: The total number of grade 9 oranges is 100,000 lb. That is, Constraint 2: The total number of grade 6 oranges is 120,000 lb. That is, Constraint 3: The average quality of juice is at least 8. That is, This gives us: Constraint 4: The average quality of juice is at least 8. That is, This gives us: Constraint 5: All variables must be nonnegative. That is, Using Excel, we can compute the optimal numbers of (checking the box Make Unconstrained Variables Non-Negative to satisfy the Constraint 5): Variables Coefficients Objective Function Constraints Constraint 1 Constraint 2 Constraint 3 Constraint 4 x11 x21 x12 x22 46666.66667 93333.33333 53333.33333 26666.66667 0.3 78000 Total 100000 120000 0 0 0.3 0.45 0.45

1 0 2 0

0 1 -1 0

1 0 0 1

0 1 0 -2

=

100000 120000 0 0

98 (4)
To make product 3 from product 1, there are 2 methods. First, we can directly process product 1 into product 3 (1 to 3), which will cost us 3 hours of labor and $2. Besides, we can process product 1 into 2, then process 2 into 3 (1 to 2 to 3), which will cost us 2+1=3 hours of labor and $1+$6=$7. Since both of the elements (i.e., labor and cost) involved in production of the first method (1 to 2) is less than those of the second one (1 to 2 to 3), we can conclude that in order to maximize profit, any rational company will reject the second method (1 to 2 to 3). Based upon the above assumption, let be the number of