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# Linear Programming : Objective Functions, Constraints, Feasible Area, Slack and Surplus Variables and Optimal Solutions

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PROBLEM 1

1. Use this graph to answer the questions.
Maximize 28X + 35Y

Subject to:

12X + 15Y < 180

15X + 10Y &#8805; 150

3X - 8Y < 0

X , Y > 0

a. What is the feasible region (I, II, III, IV, or V)?

b. Which point (A, B, C, D, or E) is optimal?

c. What is the value of the optimal solution?

d. Which constraints are binding?

e. Which slack or surplus variables is zero?

The TMA Company manufactures 19-inch color TV picture tubes in two separate locations: Location I and Location II. The monthly production capacity at Location I is 6000 tubes while that in Location II is 5000.
The picture tubes are shipped in two warehouses: Warehouse A and Warehouse B. Each month 3000 tubes must be shipped to Warehouse A and 4000 tubes to Warehouse B. The shipping costs (in dollars per picture tube) from the TMA plant to the warehouses are tabulated below:

From To Warehouse
A B
Location I \$3 \$2
Location II \$4 \$5

Find a shipping schedule that meets the above requirements while keeping the total shipping cost to a minimum. Proceed as follows:

Let x = tubes shipped from Location I to Warehouse A
y = tubes shipped from Location I to Warehouse B

a. Formulate the objective function
b. Formulate the constraints
c. Graph the constraints and indicate the feasible set.
d. Identify the optimal corner point.
e. State the optimal solution to the problem

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##### Solution Summary

Two linear programming problems are analyzed with respect to Objective Functions, Constraints, Feasible Area, Slack and Surplus Variables and Optimal Solutions.

Solution provided by:
###### 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!"
• "Thank you"
• "Thank you very much for your valuable time and assistance!"

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