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A Company Produces Three Products Which Must Be Painted, Assembled

Question 67

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:
 Product  Profit per unit  Setup cost per batch 19500210600310650\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 Operating Time per Unit\underline{\text {Operating Time per Unit}}\quad \quad \quad \quad \quad Operating\text {Operating}
 Operation  Product 1  Product 2  Product 3  Hours Available  Paint 122400 Assemble 232600 Inspection 243540\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?
Xi = amount of product i produced
Yi = 1 if product i produced, 0 otherwise
MAX: 9X1+10X2+12X3500Y1600Y2650Y3\quad 9 X _ { 1 } + 10 X _ { 2 } + 12 X _ { 3 } - 500 Y _ { 1 } - 600 Y _ { 2 } - 650 Y _ { 3 }
Subject to:
1X1+2X2+2X34002X1+3X2+2X36002X1+4X2+3X3540X1M1Y1 OR 270Y1X2M2Y2 OR 135Y2X3M3Y3OR180Y3Yi=0,1Xi0 and integer \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}
 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 (X<sub>1</sub>, X<sub>2</sub>, X<sub>3</sub>) = (270, 0, 0), what values should appear in the shaded cells in the following Excel spreadsheet? X<sub>i</sub> = amount of product i produced Y<sub>i</sub> = 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}

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