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# Theory of Linear Programming

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Sugarco can manufacture three types of candy bar. Each candy bar consists totally of sugar and chocolate. The compositions of each type of candy bar and the profit earned for each candy bar are shown in the following table.

Bar Amount of Sugar (Ounces) Amount of Chocolate (Ounces) Price (Cents)
1 1 2 3
2 1 3 7
3 1 1 5

50 oz of sugar and 100 oz of chocolate are available. After defining xi to be the number of type i candy bars manufactured, Sugarco should solve the LP

max z = 3x1 + 7x2 + 5x3
s.t. x1 + x2 + x3 50 (Sugar)
2x1 + 3x2 + x3 100 (Chocolate)
x1; x2; x3 0

After adding s1, s2 as the slack variables, the optimal tableau is as follows:

Basic x1 x2 x3 s1 s2 rhs
z 3 0 0 4 1 300
x3 1/2 0 1 3/2 -1/2 25
x2 1/2 1 0 -1/2 1/2 25

(a.) Write down the dual problem and complementary slackness conditions.

(b.) What is the maximum profit the type 1 candy bar can make to keep the current solution still optimal?

(c.) What is the allowable range of RHS of the sugar constraint?

(d.) If 60 oz sugar were available, what would be Sugarco's profit?

(e.) Suppose a type 1 candy bar used only 0.5 oz of sugar and 0.5 oz of chocolate. Should Sugarco now make type 1 candy bars?

(f.) If the available sugar decreases to 30 oz, what is the optimal solution?

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#### Solution Preview

See the attached file.
Sugarco can manufacture three types of candy bar. Each candy bar consists totally of sugar and chocolate. The compositions of each type of candy bar and the profit earned for each candy bar are shown in the following table.

Bar Amount of Sugar (Ounces) Amount of Chocolate (Ounces) Price (Cents)
1 1 2 3
2 1 3 7
3 1 1 5

50 oz of sugar and 100 oz of chocolate are available. After defining xi to be the number of type i candy bars manufactured, Sugarco should solve the LP

max z = 3x1 + 7x2 + 5x3
s.t. x1 + x2 + x3 50 (Sugar)
2x1 + 3x2 + x3 100 (Chocolate)
x1; x2; x3 0

After adding s1, s2 as the slack variables, the optimal tableau is as follows:

Basic x1 x2 x3 s1 s2 rhs
z 3 0 0 4 1 300
x3 1/2 0 1 3/2 -1/2 25
x2 1/2 1 0 -1/2 1/2 25

(a.) Write down the dual problem and complementary slackness conditions.

Min 50y1 + 100y2
s.t.
y1+2y2≥ 3
y1+3y2≥7
y1+y2≥5
y1,y2≥0

The ...

#### Solution Summary

The detailed solutions to linear programming theorem are provided in the solution.

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