5 A company manufactures two products (1 and 2). Each unit of product 1 can be sold for $15, and each unit of product 2 for $25. Each product requires raw material and two types of labor (skilled and unskilled) (see Table 29). Currently, the company has available 100 hours of skilled labor, 70 hours of unskilled labor, and 30 units of raw material. Because of marketing considerations, at least 3 units of product 2 must be produced. a Explain why the company's goal is to maximize revenue. b The relevant LP is max z = 15x + 25x2 st. 3x1 + 4x2 < 100 (Skilled labor constraint) 2x1 + 3x2 < 70 (Unskilled labor constraint) X1 + 2x2 s 30 (Raw material constraint) x2 2 3 (Product 2 constraint) X1, X2 2 0 The optimal tableau for this problem has the following row 0: z + 15s3 + Ses + (M – 5)a4 = 435 The optimal solution to the LP is z = 435, x, = 24, x2 = 3. Find and interpret the shadow price of each constraint. How much would the company be willing to pay for an ad- ditional unit of each type of labor? How much would it be willing to pay for an extra unit of raw material? C Assuming the current basis remains optimal (it does), what would the company's revenue be if 35 units of raw material were available? d With the current basis optimal, what would the com- pany's revenue be if 80 hours of skilled labor were avail- able? e With the current basis optimal, what would the com- pany's new revenue be if at least 5 units of product 2 were required? How about if at least 2 units of product 2 were required?
5 A company manufactures two products (1 and 2). Each unit of product 1 can be sold for $15, and each unit of product 2 for $25. Each product requires raw material and two types of labor (skilled and unskilled) (see Table 29). Currently, the company has available 100 hours of skilled labor, 70 hours of unskilled labor, and 30 units of raw material. Because of marketing considerations, at least 3 units of product 2 must be produced. a Explain why the company's goal is to maximize revenue. b The relevant LP is max z = 15x + 25x2 st. 3x1 + 4x2 < 100 (Skilled labor constraint) 2x1 + 3x2 < 70 (Unskilled labor constraint) X1 + 2x2 s 30 (Raw material constraint) x2 2 3 (Product 2 constraint) X1, X2 2 0 The optimal tableau for this problem has the following row 0: z + 15s3 + Ses + (M – 5)a4 = 435 The optimal solution to the LP is z = 435, x, = 24, x2 = 3. Find and interpret the shadow price of each constraint. How much would the company be willing to pay for an ad- ditional unit of each type of labor? How much would it be willing to pay for an extra unit of raw material? C Assuming the current basis remains optimal (it does), what would the company's revenue be if 35 units of raw material were available? d With the current basis optimal, what would the com- pany's revenue be if 80 hours of skilled labor were avail- able? e With the current basis optimal, what would the com- pany's new revenue be if at least 5 units of product 2 were required? How about if at least 2 units of product 2 were required?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:5 A company manufactures two products (1 and 2). Each
unit of product 1 can be sold for $15, and each unit of
product 2 for $25. Each product requires raw material and
two types of labor (skilled and unskilled) (see Table 29).
Currently, the company has available 100 hours of skilled
labor, 70 hours of unskilled labor, and 30 units of raw
material. Because of marketing considerations, at least 3
units of product 2 must be produced.
a Explain why the company's goal is to maximize
revenue.
b The relevant LP is
max z = 15x + 25x2
st.
3x1 + 4x2 < 100
(Skilled labor constraint)
2x1 + 3x2 < 70
(Unskilled labor constraint)
X1 + 2x2 s 30
(Raw material constraint)
x2 2 3
(Product 2 constraint)
X1, X2 2 0
The optimal tableau for this problem has the following row 0:
z + 15s3 + Ses + (M – 5)a4 = 435
The optimal solution to the LP is z = 435, x, = 24, x2 =
3. Find and interpret the shadow price of each constraint.
How much would the company be willing to pay for an ad-
ditional unit of each type of labor? How much would it be
willing to pay for an extra unit of raw material?
C Assuming the current basis remains optimal (it does),
what would the company's revenue be if 35 units of raw
material were available?
d With the current basis optimal, what would the com-
pany's revenue be if 80 hours of skilled labor were avail-
able?
e With the current basis optimal, what would the com-
pany's new revenue be if at least 5 units of product 2
were required? How about if at least 2 units of product
2 were required?
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