1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Anestetic [448]
3 years ago
9

A manufacturing company produces 3 different products A, B, and C. Three types of components, i.e., X, Y, and Z, are used in the

production of these products. One unit of product A requires 2 units of X, 2 units of Y, and 2 units of Z. One unit of product B requires 1 unit of X, 3 units of Y, and 2 units of Z. One unit of product C requires 1 unit of X, 2 units of Y, and 3 units of Z.Currently, the company has no existing inventory of the components. The company can purchase X at the price of $20 per unit but no more than 300 units due to the supplier's limited capacity constraint. The company can purchase Y at the price of $30 per unit without any upper limit. The company can purchase Z at the full price of $25 per unit for the first 100 units but the per unit price drops to $20 for the remaining units if any i.e., the purchase quantity in excess of 100 units). The market prices of the three products are $200 for A, $240 for B and $220 for C. The company knows that the demand for products A, B, and Care equal to 100, 80, and 90 units, respectively. Therefore, the company should not produce more than the demand for each product. However, the company incurs a per unit penalty of $40 for any unsatisfied demand. The company needs to decide the component płarchase plan and the production mix decisions to maximize its profits subject to all the business constraints described above. Formulate the company's problem as an optimization problem, i.e., providing the mathematical expressions for the decision variables, the objective function, and the constraints.
Mathematics
1 answer:
Murljashka [212]3 years ago
6 0

Answer:

Step-by-step explanation:

Using the Excel Formula:

Decision    Variable        Constraint              Constraint

A                     65                          65                         100

B                     80                          80                         80

C                     90                         90                          90

                      14100                    300                        300

= (150 *B3)+(80*B4) +(65*B5)-(100-B3+80-B4+90-B5)*90

Now, we have:

Suppose A, B, C represent the number of units for production A, B, C which is being manufactured

                             A              B                  C                Unit price

Need of X          2                 1                   1                     $20

Need of Y           2                3                  2                    $30

Need of Z           2                2                  3                    $25

Price of  

manufac -      $200          $240            $220      

turing

Now,  for manufacturing one unit of A, we require 2 units of X, 2 units of Y, 2 units of Z are required.  

Thus, the cost or unit of manufacturing of A is:

$20 (2) + $30(2) + $25(2)

$(40 + 60 + 50)

= $150

Also, the market price of A = $200

So, profit = $200 - $150 = $50/ unit of A

Again;

For manufacturing one unit of B, we require 1 unit of X, 3 units of Y, and 2 units of Z are needed and they are purchased at $20, $30, and 425 each.

So, total cost of manufacturing a unit of B is:

= $20(1) + $30(3) + $25(2)

= $(20 + 90+50)

= $160

And the market price of B = $240

Thus, profit = $240- $160  

profit = $80

For manufacturing one unit of C, we have to use 1 unit of X, 2 unit of Y, 3 units of Z are required:

SO, the total cost of manufacturing a unit of C is:

= $20 (1) + $30(2) + $25(3)

= $20 + $60 + $25

= $155

This, the profit = $220 - $155 = $65

However; In manufacturing A units of product A, B unit of product B & C units of product C.

Profit  --> 50A + 80B + 65C

This should be provided there is no penalty for under supply of there is under supply penalty for A, B, C is $40

The current demand is:

100 - A

80 - B

90 - C respectively

So, the total penalty

{(100 - A) + (80 - B) +(90 - C) } + \$40

This should be subtracted from profit.

So, we have to maximize the profit  

Z = 50A + 80B + 65C = {(100 -A) + (80 - B) + (90 - C)};

Subject to constraints;

we have the total units of X purchased can only be less than or equal to 300 due to supplies capacity

Then;

2A + B +C \le 300 due to 2A, B, C units of X are used in manufacturing A, B, C units of products A, B, C respectively.

Next; demand for A, B, C will not exceed 100, 80, 90 units.

Hence;

A \le 100

B \le 80

C \le 90

 

and A, B, C \ge 0 because they are positive quantities

The objective is:

\mathbf{Z = 50A + 80B + 65 C - (100 - A + 80 - B + 90 - C) * 40}

A, B, C \to Decision Varaibles;

Constraint are:

A \le 100 \\ \\  B \le 100 \\ \\ C \le 90 \\ \\2A + B + C \le 300 \\ \\ A,B,C \ge 0

You might be interested in
Which linear inequality is represented by the graph?
Fantom [35]

Answer:

y \leq 1/2x + 2 :)

Step-by-step explanation:

6 0
3 years ago
Read 2 more answers
A child pool is 3.8 feet wide 5.5 feet long and 1.2 fee deep, what is the volume of the pool?
Anni [7]

Answer:

25.08

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
If the perimeter of the triangle is fifteen centimeters what is the value of n?
Harrizon [31]

I am pretty sure the value of ‘n’ is 2.5!

6 0
2 years ago
Read 2 more answers
Which inequality represents this sentence ? 7 5 is greater than 6 2
Blizzard [7]
75 > 62
the way I was taught it is that the < and > are alligators and the alligator mouth always opens for the bigger number, because he is hungry
6 0
3 years ago
Similar or Like Terms are two monomials that are exactly alike except for their
grandymaker [24]
Fales for your answer
8 0
3 years ago
Read 2 more answers
Other questions:
  • The point P(8, 6) is on the terminal side of θ. Evaluate sin θ.
    12·2 answers
  • Answers this for me plzzz really quickly
    13·2 answers
  • What is 1 ft^3 to m^3
    11·1 answer
  • The diameter of a circle is 8 miles.What is the circle’s radius?
    13·1 answer
  • Solve the system of equations.
    15·2 answers
  • What is the geometric mean of 3 and 18
    5·2 answers
  • A line passes through the point (-4,6) and has a slople of 3/2. Write an equation in point-slope form for this line.
    5·1 answer
  • Find the H+ concentration of of a solution that has a pH solution of 12.24
    14·1 answer
  • Factor completely 3x2 − 11x 8. (x 4)(3x − 2) (x − 2)(3x − 4) (x 1)(3x − 8) (x − 1)(3x − 8).
    5·1 answer
  • A box contains 4 black shirts, 8 blue shirts, 4 black pants, and 10 blue pants. Determine the probability of randomly selecting
    14·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!