Linear Programming Question

[uvucs215]

I. An ice cream manufacturing company produce 2 flavored ice creams, A and B by using only one machine. To produce one liter of type A ice creams, the machine takes 3 hours and to produce 1 liter of type B ice creams the machine takes 5 hours.

a) Write a mathematical relation describing the liters of ice creams produced from each type if the machine is used 20 hours per day.

b) Due to an electrical power cut, the factory had to produce only 4 liters of type A and 2 liters of type B ice creams. How long did the machine work on that day?

c) If the Company decided to produce only type A ice creams and the total cost ? of the production is expressed as ? = 1000 + 50?, ? be the no of liters of type A. Describe the relationship between producing units and the average cost per unit.

d) Calculate the average cost per unit.


(I want the answers with steps) Thank you for helping.

 

[Deleted member 4993]

I. An ice cream manufacturing company produce 2 flavored ice creams, A and B by using only one machine. To produce one liter of type A ice creams, the machine takes 3 hours and to produce 1 liter of type B ice creams the machine takes 5 hours.

a) Write a mathematical relation describing the liters of ice creams produced from each type if the machine is used 20 hours per day.

b) Due to an electrical power cut, the factory had to produce only 4 liters of type A and 2 liters of type B ice creams. How long did the machine work on that day?

c) If the Company decided to produce only type A ice creams and the total cost ? of the production is expressed as ? = 1000 + 50?, ? be the no of liters of type A. Describe the relationship between producing units and the average cost per unit.

d) Calculate the average cost per unit.


(I want the answers with steps) Thank you for helping.

We will HELP you FIND the answers with steps.

In these types of problems, it is useful to first explicitly write - "what" you need to provide as solution. In the problem stated above:

FIND: (1),,,,,,,,,,,,,,,,,Write a mathematical relation (equation)

This problem has multiple "finds". So write those and tell us what are those.

Next - define the unknowns - explicitly. Some of the unknowns may be defined by the problem - e.g.-

C = cost production

x = the no of liters of type A

and continue.....

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

 

[uvucs215]

We will HELP you FIND the answers with steps.

In these types of problems, it is useful to first explicitly write - "what" you need to provide as solution. In the problem stated above:

FIND: (1),,,,,,,,,,,,,,,,,Write a mathematical relation (equation)

This problem has multiple "finds". So write those and tell us what are those.

Next - define the unknowns - explicitly. Some of the unknowns may be defined by the problem - e.g.-

C = cost production

x = the no of liters of type A

and continue.....

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem

In Question (a) I want to find variables, constraints and objectives

 

[Deleted member 4993]

In Question (a) I want to find variables, constraints and objectives

Correct - so let's start

a) Write a mathematical relation describing the liters of ice creams produced from each type if the machine is used 20 hours per day.

You will need to define the variables needed for writing the equation. Careful reading will tell you the variables are:
Let

Amount (Liters) of type A Ice cream produced = A liters​

​

Amount (Liters) of type B Ice cream produced =B liters​

Time to produce Type A icecream = t_a hours​

​

Time to produce Type B icecream = t_b hours​

You also know:

.......if the machine is used 20 hours per day

That will give you constraint equation function.

Please write the constraint equation function using t_a and t_b .

 

[uvucs215]

Correct - so let's start


You will need to define the variables needed for writing the equation. Careful reading will tell you the variables are:
Let

Amount (Liters) of type A Ice cream produced = A liters​

​

Amount (Liters) of type B Ice cream produced =B liters​

Time to produce Type A icecream = t_a hours​

​

Time to produce Type B icecream = t_b hours​

You also know:


That will give you constraint equation.

Please write the constraint equation using t_a and t_b .

t_a + t_b <= 20

 

[Deleted member 4993]

t_a + t_b <= 20

Correct ..... now write the relation (equation) for 1(a) using:

o produce one liter of type A ice creams, the machine takes 3 hours and to produce 1 liter of type B ice creams the machine takes 5 hours.

First write t_a & t_b in terms of A and B and try to make a relation using the constraint

 

[uvucs215]

Correct ..... now write the relation (equation) for 1(a) using:

First write t_a & t_b in terms of A and B and try to make a relation using the constraint

A >= t_a + B >= t_b <= 20

Is this correct?

 

[Deleted member 4993]

A >= t_a + B >= t_b <= 20

Is this correct?

As far as I can tell - NO!

What is "unit" of t_a and t_b? What is "unit" of A and B? The relationship above must have "uniform" unit (dimension).

This relationship must be compatible with your constraint equation too (response # 5)!

 

[uvucs215]

As far as I can tell - NO!

What is "unit" of t_a and t_b? What is "unit" of A and B? The relationship above must have "uniform" unit (dimension).

This relationship must be compatible with your constraint equation too (response # 5)!

3<= t_a + 5<= t_b <= 20

I'm sorry. it is bit difficult me to understand.

 

[Deleted member 4993]

3<= t_a + 5<= t_b <= 20

I'm sorry. it is bit difficult me to understand.

How did you arrive at the term t_a +5 ? What is 5 coming from?

 

[uvucs215]

How did you arrive at the term t_a +5 ? What is 5 coming from?

Ah I put it like this

(3 <= t_a) + (5 <= t_b) <= 20

5 is the hours - type B ice cream produce time

 

[Deleted member 4993]

I. An ice cream manufacturing company produce 2 flavored ice creams, A and B by using only one machine. To produce one liter of type A ice creams, the machine takes 3 hours and to produce 1 liter of type B ice creams the machine takes 5 hours.

a) Write a mathematical relation describing the liters of ice creams produced from each type if the machine is used 20 hours per day.

b) Due to an electrical power cut, the factory had to produce only 4 liters of type A and 2 liters of type B ice creams. How long did the machine work on that day?

c) If the Company decided to produce only type A ice creams and the total cost ? of the production is expressed as ? = 1000 + 50?, ? be the no of liters of type A. Describe the relationship between producing units and the average cost per unit.

d) Calculate the average cost per unit.


(I want the answers with steps) Thank you for helping.

I am going to solve a similar (but not same) problem for part a:

I. An ice cream manufacturing company produce 2 flavored ice creams, V(anilla) and S(trawberries) by using only one machine. To produce one liter of type V ice creams, the machine takes 2 hours and to produce 1 liter of type S ice creams the machine takes 4 hours.

a) Write a mathematical relation describing the liters of ice creams produced from each type if the machine is used 18 hours per day.

Let
Amount (Liters) of type V Ice cream produced = V liters
Amount (Liters) of type S Ice cream produced =S liters

Time to produce Type V icecream = t_v hours
Time to produce Type S icecream = t_s hours

One of the ways to obtain the relationship:

t_v + t_s = 18 .......... t_s = 18 - t_v .....................................(1)
We know
to produce one liter of type V ice creams, the machine takes 2 hours and to produce 1 liter of type S ice creams the machine takes 4 hours.

then

t_v / 2 = V .......(2).............and ....................

t_S / 4 =

(18 - t_v)/4 = S .........(3).................using (1)

Using (2) in (3)

(18 - 2 * V)/4 = S

2*V + 4*S = 18

V + 2*S = 9..............................................desired relationship

Follow the same (similar) steps to derive the desired relationship for your given assignment.

 

[uvucs215]

I am going to solve a similar (but not same) problem for part a:

I. An ice cream manufacturing company produce 2 flavored ice creams, V(anilla) and S(trawberries) by using only one machine. To produce one liter of type V ice creams, the machine takes 2 hours and to produce 1 liter of type S ice creams the machine takes 4 hours.

a) Write a mathematical relation describing the liters of ice creams produced from each type if the machine is used 18 hours per day.



One of the ways to obtain the relationship:

t_v + t_s = 18 .......... t_s = 18 - t_v .....................................(1)
We know
to produce one liter of type V ice creams, the machine takes 2 hours and to produce 1 liter of type S ice creams the machine takes 4 hours.

then

t_v / 2 = V .......(2).............and ....................

t_S / 4 =

(18 - t_v)/4 = S .........(3).................using (1)

Using (2) in (3)

(18 - 2 * V)/4 = S

2*V + 4*S = 18

V + 2*S = 9..............................................desired relationship

Follow the same (similar) steps to derive the desired relationship for your given assignment.

ohh I got it.

t_a + t_b = 20

t_b = 20 - t_a ------------ (1)

next,

t_a / 3 = A ----------- (2)

t_b / 5 = (20 - t_a) / 5 = B ------------ (3)

Using (2) in (3)

(20 - 3*A) / 5 = B

3*A + 5*B = 20

A + 1.67*B = 6.67

correct?

 

[Deleted member 4993]

ohh I got it.

t_a + t_b = 20

t_b = 20 - t_a ------------ (1)

next,

t_a / 3 = A ----------- (2)

t_b / 5 = (20 - t_a) / 5 = B ------------ (3)

Using (2) in (3)

(20 - 3*A) / 5 = B

3*A + 5*B = 20

A + 1.67*B = 6.67

correct?

CORRECT

Now start on the rest of the problem - parts (b), (c) & (d)....

 

[uvucs215]

CORRECT

Now start on the rest of the problem - parts (b), (c) & (d)....

part (b)

Due to an electrical power cut, the factory had to produce only 4 liters of type A and 2 liters of type B ice cream. How long did the machine work on that day

Answer:

4*A <= t_a
2*B <= t_b

3 * (4A) <= t_a (to produce 1 liter of type A ice cream, machine takes 3 hrs)
5 * (2B) <= t_b (to produce 1 liter of type B ice cream, machine takes 5 hrs)

3 * 4A = t_a
5 * 2B = t_b

t_a = 12
t_b = 10

Total hours = 12 + 10
= 22 hours


part (c)

If the company decided to produce only type A ice cream and the total cost C of the production is expressed as C = 1000 + 50x , x be the no of liters of type A. Describe the relationship between producing units and the average cost per unit.

Answer:

C = 1000 + 50x
Q = number of liters

Average cost = C / Q

= (1000 + 50x) / x

= (1000 / x) + 50

part (d)

Calculate the average cost per unit

Answer:

Average cost = (1000 / x) + 50

= (1000 / 1) + 50

= 1000 + 50

= 1050




Is this correct sir?

 

[Deleted member 4993]

I. An ice cream manufacturing company produce 2 flavored ice creams, A and B by using only one machine. To produce one liter of type A ice creams, the machine takes 3 hours and to produce 1 liter of type B ice creams the machine takes 5 hours.

a) Write a mathematical relation describing the liters of ice creams produced from each type if the machine is used 20 hours per day.

b) Due to an electrical power cut, the factory had to produce only 4 liters of type A and 2 liters of type B ice creams. How long did the machine work on that day?

c) If the Company decided to produce only type A ice creams and the total cost ? of the production is expressed as ? = 1000 + 50?, ? be the no of liters of type A. Describe the relationship between producing units and the average cost per unit.

d) Calculate the average cost per unit.


(I want the answers with steps) Thank you for helping.

I'll solve part (b), part (c) and part (c) for you and you can compare, check and decide.

b)Time required to make 4 liters of type A = 4 (L) * 3 (H/L) = 12 H

Time required to make 2 liters of type B = 2 (L) * 5 (H/L) = 10 H

The machine worked for (12+10=) 22 Hrs. that day.

c) The fixed cost of production is 1000. Thus the average cost per unit (C/x) will decrease with increase of number of units ( 'x' )

d) average cost per unit = C/x = (1000/x) + 50

Now check your answers......

 

[uvucs215]

I'll solve part (b), part (c) and part (c) for you and you can compare, check and decide.

b)Time required to make 4 liters of type A = 4 (L) * 3 (H/L) = 12 H

Time required to make 2 liters of type B = 2 (L) * 5 (H/L) = 10 H

The machine worked for (12+10=) 22 Hrs. that day.

c) The fixed cost of production is 1000. Thus the average cost per unit (C/x) will decrease with increase of number of units ( 'x' )

d) average cost per unit = C/x = (1000/x) + 50

Now check your answers......

Ohh.. it's correct.

Thank you so much for your help. Thank you so much for making free platform like this to get help. I really appreciate your help. Stay safe. Good luck for you future.

 

[Deleted member 4993]

Check your answer for part (d). You have calculated the cost of 1 unit when the total production is 1.

The question did NOT ask for that.

 

[uvucs215]

Check your answer for part (d). You have calculated the cost of 1 unit when the total production is 1.

The question did NOT ask for that.

So, for part (d)

average per unit answer 1050 is wrong?