A Company Produces Three Products Which Must Be Painted, Assembled

Essay

A company produces three products which must be painted, assembled, and inspected. The machinery must be cleaned and adjusted before each batch is produced. They want to maximize their profits for the amount of operating time they have. The unit profit and setup cost per batch are:
$\begin{array}{ccc}\text { Product } & \text { Profit per unit } & \text { Setup cost per batch } \\\hline 1 & 9 & 500 \\2 & 10 & 600 \\3 & 10 & 650\end{array}$
The operation time per unit and total operating hours available are:
$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\underline{\text {Operating Time per Unit}}$$\quad$$\quad$$\quad$$\quad$$\quad$$\text {Operating}$
$\begin{array}{lcccc}\text { Operation } & \text { Product 1 } & \text { Product 2 } & \text { Product 3 } & \text { Hours Available } \\\hline \text { Paint } & 1 & 2 & 2 & 400 \\\text { Assemble } & 2 & 3 & 2 & 600 \\\text { Inspection } & 2 & 4 & 3 & 540\end{array}$

Based on this ILP formulation of the problem and the optimal solution (X₁, X₂, X₃) = (270, 0, 0), what values should appear in the shaded cells in the following Excel spreadsheet?
X_i = amount of product i produced
Y_i = 1 if product i produced, 0 otherwise
MAX: $\quad 9 X _ { 1 } + 10 X _ { 2 } + 12 X _ { 3 } - 500 Y _ { 1 } - 600 Y _ { 2 } - 650 Y _ { 3 }$
Subject to:
$\begin{array}{l}1 \mathrm{X}_{1}+2 \mathrm{X}_{2}+2 \mathrm{X}_{3} \leq 400 \\2 \mathrm{X}_{1}+3 \mathrm{X}_{2}+2 \mathrm{X}_{3} \leq 600 \\2 \mathrm{X}_{1}+4 \mathrm{X}_{2}+3 \mathrm{X}_{3} \leq 540 \\\mathrm{X}_{1} \leq \mathrm{M}_{1} \mathrm{Y}_{1} \text { OR } 270 \mathrm{Y}_{1} \\\mathrm{X}_{2} \leq \mathrm{M}_{2} \mathrm{Y}_{2} \text { OR } 135 \mathrm{Y}_{2} \\\mathrm{X}_{3} \leq \mathrm{M}_{3} \mathrm{Y}_{3} \mathrm{OR} 180 \mathrm{Y}_{3} \\\mathrm{Y}_{\mathrm{i}}=0,1 \\\mathrm{X}_{\mathrm{i}} \geq 0 \text { and integer }\end{array}$