A manufacturer has the capability to produce both chairs and tables. Both products use the same materials (wood, nails and paint) and both have a setup cost ($100 for chairs, $200 for tables). The firm earns a profit of $20 per chair and $65 per table and can sell as many of each as it can produce. The daily supply of wood, nails and paint is limited. To manage the decision-making process, an analyst has formulated the following linear programming model:
Max 20x1 + 65x2 - 100y1 - 200y2
s.t. 5x1 + 10x2 ? 100 {Constraint 1}
20x1 + 50x2 ? 250 {Constraint 2}
1x1 + 1.5x2 ? 10 {Constraint 3}
My1 ? x1 {Constraint 4}
My2 ? x2 {Constraint 5} $$y _ { i } = \left\{ \begin{array} { l }
1 , \text { if product } j \text { is produced } \
0 , \text { otherwise }
\end{array} \right.$$
Which of the following would be a reasonable value for the variable "M"?
A) 100
B) 10
C) 1
D) 0.1
E) 0.01

Question

A manufacturer has the capability to produce both chairs and tables. Both products use the same materials (wood, nails and paint) and both have a setup cost ($100 for chairs, $200 for tables). The firm earns a profit of $20 per chair and $65 per table and can sell as many of each as it can produce. The daily supply of wood, nails and paint is limited. To manage the decision-making process, an analyst has formulated the following linear programming model:
Max 20x1 + 65x2 - 100y1 - 200y2
s.t. 5x1 + 10x2 ? 100 {Constraint 1}
20x1 + 50x2 ? 250 {Constraint 2}
1x1 + 1.5x2 ? 10 {Constraint 3}
My1 ? x1 {Constraint 4}
My2 ? x2 {Constraint 5} $$y _ { i } = \left\{ \begin{array} { l } 
1 , \text { if product } j \text { is produced } \
0 , \text { otherwise }
\end{array} \right.$$
Which of the following would be a reasonable value for the variable "M"?
A) 100
B) 10
C) 1
D) 0.1
E) 0.01

Quizplus · Accepted Answer

Constraints 4 and 5 are designed to ensure that $y_1$ and $y_2$ are set to 1 if any chairs (x1) or tables (x2) are produced, respectively. The value of "M" must be large enough to be greater than any possible production quantity of chairs or tables, ensuring that the constraints are only active when $y_1$ or $y_2$ are supposed to be 1. Given the context and the constraints on resources, a value of 100 is reasonable because it is sufficiently large to cover the maximum production quantities of chairs or tables, ensuring the binary conditions for $y_1$ and $y_2$ are correctly implemented.

A Manufacturer Has the Capability to Produce Both Chairs and Tables