Chapter 3, Problem 35RP

Explanation of Solution

 Formulation of Linear Programming (LP):

 Let “x₁” be the type 1 layout, “x₂” be type 1 layout, and “x₃” be type 3 layout.

 The objective is to maximize the profit.

 $\begin{array}{c}z=\left[\begin{array}{l}\left(\text{net profit}\right)\left(\text{type 1 layout}\right)+\left(\text{net profit}\right)\left(\text{type 2 layout}\right)+\\ \left(\text{net profit}\right)\left(\text{type 3 layout}\right)\end{array}\right]\\ =50x_1+30x_2+60x_3\end{array}$

 Therefore, the objective function is,

 Maximize, $z=50x_1+30x_2+60x_3$

 Constraint 1:

 At most, 1000 tulips are available.

 $\begin{array}{c}\left[\begin{array}{l}\left(\text{tulips used}\right)\left(\text{type 1 layout}\right)+\left(\text{tulips used}\right)\left(\text{type 2 layout}\right)+\\ \left(\text{tulips used}\right)\left(\text{type 3 layout}\right)\end{array}\right]\le 1000\\ 30x_1+10x_2+20x_3\le 1000\end{array}$

 Constraint 2:

 At most, 800 daffodils are available