A bakery produces both pies and cakes. Both products use the same materials (flour, sugar and eggs) and both have a setup cost ($100 for cakes, $200 for pies). The baker earns a profit of $10 per cake and $12 per pie and can sell as many of each as it can produce. The daily supply of flour, sugar and eggs is limited. To manage the decision-making process, an analyst has formulated the following linear programming model (assume that it is possible to produce fractional pies and cakes for this example):
Max 10x1 + 12x2 - 100y1 - 200y2
s.t. 5x1 + 10x2 ? 1000 {Constraint 1}
2x1 + 5x2 ? 2500 {Constraint 2}
2x1 + 1x2 ? 300 {Constraint 3}
My1 ? x1 {Constraint 4}
My2 ? x2 {Constraint 5} $$y _ { i } = \left\{ \begin{array} { l }
1 , \text { if product } j \text { is produced } \
0 , \text { ot herwise }
\end{array} \right.$$
Which of the following would be a reasonable value for the variable "M"?
A) 0.1
B) 1
C) 10
D) 100
E) 1,000

Question

A bakery produces both pies and cakes. Both products use the same materials (flour, sugar and eggs) and both have a setup cost ($100 for cakes, $200 for pies). The baker earns a profit of $10 per cake and $12 per pie and can sell as many of each as it can produce. The daily supply of flour, sugar and eggs is limited. To manage the decision-making process, an analyst has formulated the following linear programming model (assume that it is possible to produce fractional pies and cakes for this example):
Max 10x1 + 12x2 - 100y1 - 200y2
s.t. 5x1 + 10x2 ? 1000 {Constraint 1}
2x1 + 5x2 ? 2500 {Constraint 2}
2x1 + 1x2 ? 300 {Constraint 3}
My1 ? x1 {Constraint 4}
My2 ? x2 {Constraint 5} $$y _ { i } = \left\{ \begin{array} { l } 
1 , \text { if product } j \text { is produced } \
0 , \text { ot herwise }
\end{array} \right.$$
Which of the following would be a reasonable value for the variable "M"?
A) 0.1
B) 1
C) 10
D) 100
E) 1,000

Quizplus · Accepted Answer

The variable "M" should be a large number to ensure that the constraints My1 ≥ x1 and My2 ≥ x2 are not binding when y1 or y2 is 0. A value like 1,000 is large enough to satisfy this condition.

A Bakery Produces Both Pies and Cakes yi={1, if product j is produced 0, ot herwise y _ { i } = \left\{ \begin{array} { l } 1 , \text { if product } j \text { is produced } \\0 , \text { ot herwise }\end{array} \right.yi​={1, if product j is produced 0, ot herwise ​

A Bakery Produces Both Pies and Cakes $y _ { i } = \left\{ \begin{array} { l } 1 , \text { if product } j \text { is produced } \\0 , \text { ot herwise }\end{array} \right.$