The Following Linear Programming Problem Has Been Solved by the Management

question 2

Essay

The following linear programming problem has been solved by The Management Scientist.Use the output to answer the questions.
LINEAR PROGRAMMING PROBLEM
MAX 25X1+30X2+15X3
S.T.
1)4X1+5X2+8X3<1200
2)9X1+15X2+3X3<1500
OPTIMAL SOLUTION
Objective Function Value = 4700.000 $$\begin{array} { c c c } \text { Variable } & \text { Value } & \text { Reduced Cost } \\\text { X1 } & 140.000 & 0.000 \\\text { X2 } & 0.000 & 10.000 \\\text { X3 } & 80.000 & 0.000\end{array}$$ $$\begin{array} { c c c } \text { Constraint } & \text { Slack/Surplus } & \text { Dual Price } \\1 & 0.000 & 1.000 \\2 & 0.000 & 2.333\end{array}$$ OBJECTIVE COEFFICIENT RANGES $$\begin{array} { c c c c } \text { Variable } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\\text { X1 } & 19.286 & 25.000 & 45.000 \\\text { X2 } & \text { No Lower Limit } & 30.000 & 40.000 \\\text { X3 } & 8.333 & 15.000 & 50.000\end{array}$$ RIGHT HAND SIDE RANGES $$\begin{array} { c c c c } \text { Constraint } & \text { Lower Limit } & \text { Current Value } & \text { Upper Limit } \\1 & 666.667 & 1200.000 & 4000.000 \\2 & 450.000 & 1500.000 & 2700.000\end{array}$$
a.Give the complete optimal solution.
b.Which constraints are binding?
c.What is the dual price for the second constraint? What interpretation does this have?
d.Over what range can the objective function coefficient of x₂ vary before a new solution point becomes optimal?
e.By how much can the amount of resource 2 decrease before the dual price will change?f. What would happen if the first constraint's right-hand side increased by 700 and the second's decreased by 350?

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Definitions:

Artificiality

The condition or quality of being artificial or synthetic, as opposed to naturally occurring.

Real-world Environments

Settings or situations that occur in everyday life, outside of controlled or experimental conditions.

Random Assignment

A method used in experiments to evenly distribute participants across various treatment groups, ensuring each group is comparable at the start.

Experimental Group

The group in an experiment that receives the variable being tested, allowing researchers to determine the effect of that variable.