Chapter 3.8, Problem 11P

Explanation of Solution

 Formulation of a Linear Programming (LP) to determine the cheapest way of producing today’s batch of Rozac:

 In the given table, “A” be the pounds of ingredient “A” used, “B” be the pounds of ingredient “B” used, and “C” be the pounds of ingredient “C” used.

 Let “x_i” be the pounds of chemical “i” used.

 The objective is to minimize the cost of production.

 $\begin{array}{c}z=\left(\text{cost of chemical 1}\right)\left(\text{pounds of chemical 1}\right)+\left(\text{cost of chemical 2}\right)\left(\text{pounds of chemical 2}\right)+\\ \left(\text{cost of chemical 3}\right)\left(\text{pounds of chemical 3}\right)+\left(\text{cost of chemical 4}\right)\left(\text{pounds of chemical 4}\right)\\ =8x_1+10x_2+11x_3+14x_4\end{array}$

 Therefore, the objective function is,

 $z=8x_1+10x_2+11x_3+14x_4$

 Constraint 1 to 3:

 Minimum ingredient A, B, C requirements for the drug.

 $\begin{array}{c}\left(\text{pounds of ingredient A used}\right)\ge \left[\begin{array}{l}\left(\text{at least 8\% of drug must contain A}\right)\times \\ \left(\text{total production of drug}\right)\end{array}\right]\\ A\ge \left(0.08\right)\left(1000\right)\\ A\ge 80\end{array}$

 $\begin{array}{c}\left(\text{pounds of ingredient A used}\right)\ge \left[\begin{array}{l}\left(\text{at least 4\% of drug must contain B}\right)\times \\ \left(\text{total production of drug}\right)\end{array}\right]\\ B\ge (0\end{array}$