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Problems

A-45

5. Given is the following simplex tableau for a linear programming problem:

A x

1 1

2 6

0 1

1/2 1/2

-1 0

1/2 -1>2

1 0

z

j

24 + 2M

4

M>2 + 2

-M

M>2 - 2

M

z -c

j

j

0

M>2 - 4

-M

M>2 - 2

0

Basic

4

6

0

0

M

j c

Variables

Quantity

x1

x2

s1

s2

A1

M 4

  • a.

    Is this a maximization or a minimization problem? Why?

  • b.

    What are the values of the decision variables in this tableau?

  • c.

    Were any of the constraints in this problem originally equations? Why?

  • d.

    What is the value of s2 in this tableau?

  • e.

    Is this solution optimal? Why? If the solution is not optimal, complete the next iteration

(tableau) and indicate if it is optimal.

6. Following is a simplex tableau for a linear programming problem:

10

Quantity

x1

5

1

4

0

15

0

-4M + 50

10

0

1/2

-1>2

0

0

1

0

0

1

7/2

1/2

1

0

-M + 5

-5

0

-M

M

5

0

0

5

0

0

-M

x2

s1

s2

A2

z j

c j - z j

c j

Basic Variables

10

  • -

    M

0

x A s2 1 2

a. Is this a maximization or a minimization problem? Why? b. What is the value of x2 in this tableau? c . D o e s t h e f a c t t h a t h a s a v a l u e e q u a l t o 0 i n t h i s t a b l e a u m e a n t h a t m u l t i p l e c j - z j x 1

optimal solutions exist? Why? d . W h a t d o e s t h e v a l u e f o r t h e c o l u m n m e a n ? s 1 c j - e. Is this solution optimal? Why? If not, solve this problem and indicate if multiple optimal z j

solutions exist.

7. The Munchies Cereal Company makes a cereal from several ingredients. Two of the ingredients, oats and rice, provide vitamins A and B. The company wants to know how many ounces of oats and rice it should include in each box of cereal to meet the minimum requirements of 48 milligrams of vitamin A and 12 milligrams of vitamin B while minimizing cost. An ounce of oats contributes 8 milligrams of vitamin A and 1 milligram of vitamin B, whereas an ounce of rice contributes

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