Statistics- What is the standard form of constraint

1)
Put the following LP in standard
form.

Let: X_i
= the number of color i shoes to produce per day

I
= 1 = White, 2 = Grey, 3 = Brown

Maximize Profit Z = 3X₁+4X₂+4.5X₃

St: 2X₁+16X₂+20X₃
≤ 5,000 (1)

4X₁+9X₂≤
4,800 (2)

-8X₃≤-2,500 (3)

2X₁≤X₂+X₃ (4)

X₁,
X_2,X₃≥0

a)
The first constraint is developed
because there are only 5000 grommets available per day and white shoes use 2,
grey use 16 and brown use 20. What would
be the definition of the slack variable added to this constraint?

b)
What is the standard form of
constraint (3)?

c)
What is the standard form of
constraint (4)?

2)
Find the optimal solution for
the following linear program using the graphical method.

Let: X₁
= the number of hats to produce per day

X₂ = the number of shirts to produce per day

Maximize Profit Z = 2X₁+6X₂

St: 8X₁+10X₂≤
80 (1)

4X₁+8X₂≥
32 (2)

5X₁≤20
(3)

X₁, X₂≥0

a)
What is the optimal number of
hats and shirts to produce? What is the
maximum profit with these constraints?

b)
How many extreme points
(possible optimal points) are there in the feasible region?

c)
Are there any redundant
constraints and if so what are they?

d)
Which constraints are binding
and what are their slack and surplus values?

3)
A manufacturing manager has
three different sewing machines for making t-shirts. The demand per day is 135 t-shirts. The parameters of each machine are given
below. A shirt that is started on one
machine must be finished on that machine. In other words, they can only make a
whole shirt.

Sewing Machine	Set-up Cost per Day ($)	Cost per T-shirt	Capacity per Day
1	30	$0.21	95
2	32	$0.17	65
3	35	$0.16	90

Write a linear
programming formulation to help the manager decide how many t-shirts to produce
on each machine per day. (Hint: You need
to decide if the machine will be used or not.
Then, if it is used, how many shirts will be made on that machine.)

a)
How many decision variables are
needed for this problem?

b)
Write out the objective for the
problem.

c)
Write the constraint for sewing
machine 1.

4)
Your manager has asked you to
determine which projects your consulting group should work on this year to make
the most profit. The projects are listed
below with the profit and cost for each.

Project #	Net Profit ($)	Cost ($)
1	10,000	2,000
2	50,000	10,000
3	12,000	5,000
4	24,000	6,000
5	45,000	8,000

y_i
= 1 if project i is completed and 0 if not where i = 1,2,3,4,5

a)
The budget for the year is
$20,000. Write a constraint for this.

b)
Also, projects 1 and 2 are for
the same company but at different involvement levels so they are mutually exclusive
(you cannot choose to do both). Write a
constraint for this.

c)
Project 4 can only be done if
project 3 is also selected. Write a
constraint for this.

d)
At most, they want to do 2 of
projects 2, 4, and 5. Write a constraint
for this.

Multichoice
Questions

Note- the Multichoice are the
same as the above questions so u dont have to work them. Just click on the
CORRECT answer.

1. The first constraint
is developed because there are only 5000 grommets available per day and the
white shoes use 2, grey use 16 and brown use 20. What would be the
definition of the slack variable added to this constraint?

Answer

		The slack variable is the amount beyond the required 5000 grommets used each day.
		The slack variable is the amount of grommets used each day.
		The slack variable is the amount of grommets used on white shoes per day.
		The slack variable is the amount of grommets that are leftover each day after the shoes are produced.

Question 2

1.

What is the standard form of constraint (3)?

Answer

		8×3 – s3 = 2,500
		8×3 + s3 = 2,500
		-8×3 + s3 = -2,500
		-8×3 – s3 = -2,500

Question 3

1.

What is the standard form of constraint (4)?

Answer

		2×1 + s4 = x2 + x3
		2×1 – s4 = x2 + x3
		2×1 – x2 – x3 – s4 = 0
		2×1 – x2 – x3 + s4 = 0

Question 4

What is the optimal number of hats and shirts to produce?
What is the maximum profit with these constraints?

Answer

		Hats = 4, Shirts = 2, Profit = 20
		Hats = 0, Shirts = 8, Profit = 48
		Hats = 0, Shirts = 4, Profit = 24
		Hats = 8, Shirts = 0, Profit = 48

Question
5

How many extreme points (possible optimal points) are there in the
feasible region?

Answer

		4
		5
		3
		2

Question 6

Are there any redundant constraints and if so what are they?

Answer

		Yes, constraint (1) is redundent
		Yes, constraint (3) is redundant.
		No, there are no redundant constraints.
		Yes, constraint (2) is redundant.

Question 7

Which constraints are binding and what are their slack and/or
surplus values?

Answer

		Constraint (1) is binding and has a slack value of 0.
		Constraint (1) is binding and has a slack value of 40.
		Constraints (2) and (3) are binding and have a surplus and slack value of 32 and 20.
		Constraint (2) is binding and has a slack value of 0.

Question 8

How many decision variables are needed for this problem?

Answer

		2
		6
		3
		9

Question 9

Write out the objective for the problem.

Answer

		Min C = (30+32+35)y + (0.21+0.17+0.16)x Where y is 1 if the machine is turned on and 0 if not and x is the number of shirts to make per day
		Min C=30y1 + 32y2 + 35y3 + 0.21×1 + 0.17×2 + 0.16×3 where yi is 1 if machine I is turned on and 0 if not and xi is the number of shirts made on machine i per day.
		Min C = 30*0.21×1 + 32*0.17×2 + 35*0.16×3 where xi is the number of shirts made on machine i per day
		Min C = 0.21×1 + 0.17×2 + 0.16×3 where xi is the number of shirts made on machine i per day

Question 10

Write out the constrain for sewing machine 1.

Answer

		x1* y1 <= 95 where x1 is the number of shirts to make on machine 1 per day and y1 is 1 if machine 1 is turned on and 0 if not.
		x1<= 95 where x1 is the number of shirts to make on machine 1 per day.
		x1 <= 95y1 where x1 is the number of shirts to make on machine 1 per day and y1 is 1 if machine 1 is turned on and 0 if not.
		x*y <= 95 where x is the of shirts to make per day and y is 1 if the machine is turned on and 0 if not.

Question 11

The budget for the year is $20,000. Write a constraint for
this.

Answer

		2,000y1 + 10,000y2 + 5,000y3 + 6,000y4 + 8,000y5 <= 20,000
		y1 + y2 + y3 + y4 + y5 <= 20,000
		y <= 20,000
		10,000y1 + 50,000y2 + 12,000y3 + 24,000y4 + 45,000y5 <= 20,000

Question 12

Also, projects 1 and 2 are for the same company but at different
involvement levels so there are mutually exclusive (you cannot choose to do
both). Write a constraint for this.

Answer

		y1 + y2 >= 1
		y1 + y2 <=1
		y1 – y2 <= 1
		y1 + y2 = 1

Question 13

Project 4 can only be done if project 3 is also selected.
Write a constraint for this.

Answer

		y4 + y3 <=1
		y4 + y3 <= 2
		y4 >= y3
		y4 <= y3

Question 14

At most, they want to do 2 of projects 2, 4, and 5. Write a
constraint for this.

Answer

		y2 + y4 + y5 >= 2
		y2 + y4 + y5 = 2
		y2 – y4 – y5 <=2
		y2 + y4 + y5 <=2