# Linear Programming : Simplex Method

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1. Consider the following maximum problem in standard form:

Maximize Z = 8X1 + 2X2 + 3X3

subject to the constraints

X1 + 3X2 + 2X3 < 10

4X1 + 2X2 + 3X3 < 8

X1 > 0, X2 > 0, X3 > 0

(a) Rewrite the two constraints as equations by adding slack variables S1 and S2.

(b) Set up the initial simplex tableau for this problem.

In problems 2 -4, each tableaux represents a step in the solution of a maximization problem in standard form. Determine if the tableaux:

(i) is the final tableaux

(ii) requires additional pivoting

(iii) indicates no solution to the problem

If the answer is (i), write down the solution; if the answer is (ii), identify the pivot element.

2.

3.

4.

5. A brewery manufactures three types of beer - lite, regular, and dark. Each vat of lite beer requires 6 bags of barley, 1 bag of sugar and 1 bag of hops. Each vat of regular beer requires 4 bags of barley, 3 bag of sugar and 1 bag of hops. Each vat of dark beer requires 2 bags of barley, 2 bag of sugar and 4 bag of hops. Each day the brewery has 800 bags of barley, 600 bag of sugar and 300 bag of hops. The brewery realizes a profit of $10 per vat of lite beer, $20 per vat of regular beer, and $30 per vat of dark beer. For this linear programming problem:

(a) What are the decision variables?

(b) What is the objective function?

(c) What are the constraints?

E.C. Solve the linear program in problem #5.

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LP problems are solved using the simplex method.

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1. Consider the following maximum problem in standard form:

Maximize Z = 8X1 + 2X2 + 3X3

subject to the constraints

X1 + 3X2 + 2X3 < 10

4X1 + 2X2 + 3X3 < 8

X1 > 0, X2 > 0, X3 > 0

(a) Rewrite the two constraints as equations by adding slack variables S1 and S2.

X1 + 3X2 + 2X3 +S1 = 10

4X1 + 2X2 + 3X3 +S2 = 8

(b) Set up the initial simplex tableau for this problem.

|-----------------------------------------|-----

| -8 -2 -3 0 0 1 | 0

In problems 2 -4, each tableaux represents a step in the solution of a maximization problem in standard form. Determine if the tableaux:

(i) is the final tableaux

(ii) requires additional pivoting

(iii) indicates no solution to the problem

If the answer ...

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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