Question

2.A finished product must weight exactly 150 grams. The two raw materials used inmanufacturing the product are A, with a cost of birr 2 per unit and B with a cost of birr 8per unit. At least 14 units of B and not more than 20 units of A must be used. Each unit ofA and B weights 5 and 10 grams respectively. How much of each type of raw materialshould be used for each unit of the final product in order to minimize the cost? Use thesimplex method

Answer 1

Answer:

PROBLEM 01 – 0009: In a manufacturing process, the final product has a

requirement that it must weigh exactly 150 pounds. The two

raw materials used are A, with a cost of $4 per unit and B,

with a cost of $8 per unit. At least 14 units of B and no more

than 20 units of A must be used. Each unit of A weighs

5 pounds; each unit of B weighs 10 pounds.

How much of each type of raw material should be used for

each unit of final product to minimize cost?

Solution: The objective function is:

C = 4x1 + 8x2. (1)

The constraints are:

5x1 + 10x2 = 150

x1 ≤ 20

x2 ≥ 14 (2)

x1 ≥ 0.

Take the graphical approach to this linear programming problem. The

constraints, (2) are graphed in the figure.

Since the pertinent region lies within 0 ≤ x1 ≤ 20, x2 ≥ 14, and on

5x1 + 10x2 = 150, two solutions can be immediately found, points a and b

where a = (0, 15) and b = (2, 14).

Solution 1

Solution 2

Raw material A, (x1)

0

2

Raw material B, (x2)

15

14

Total cost, 4x1 + 8x2

120

120

This is an example of a problem having multiple solutions. In such

problems, two or more corner points have the same optimum value.

Answer 2

Answer: Minimum cost is 116 when we take 2 units of A and 14 units of B.

Explanation:

Let the units of A be $x$ and units of B be $y$

Cost of 1 unit of A  =  2

Total cost of A  =  $2x$

Cost of 1 unit of B  =  8

Total cost of B  =  $8y$

Net cost  $= 2x + 8y$

Mass of 1 unit of A  =  5

Total mass of A  =  $5x$

Mass of 1 unit of B  =  10

Total mass of B  =  $10y$

Net mass  $= 5x + 10y$

Requirement :

Minimize net cost ⇒ Minimize 2x + 8y

Constraints :

• Net mass = 150
  $5x + 10y = 150$
• Units of A not more than 20
  $x\leq 20$
• Units of B not less than 14
  $y \geq 14$
• Number of units of A and B are non-negative
  $x,y \geq 0$

Corner points are the possible solutions.

There are 2 corner points :

1. x = 0 , y = 15
2. x = 2 , y =  14

Cost  for case 1  $= 2(0) + 8(15) = 120$

Cost  for case 2  $= 2(2) + 8(14) = 116$

Minimum cost is 116 when we take 2 units of A and 14 units of B.

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