5.3 Q-6

Question Statement

Maximize the objective function: $z = 3x + y$

Subject to the following constraints:

$$3x + 5y \geq 15$$ $$x + 3y \leq 9$$ $$x \geq 0$$ $$y \geq 0$$

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Background and Explanation

This problem involves linear programming where we maximize an objective function subject to constraints. The solution involves:

1. Graphing the constraints to find the feasible region.
2. Identifying the corner points of the feasible region.
3. Evaluating the objective function at these points to find the optimal solution.

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Solution

1. Graph the constraints:

   • Constraint 1: $3x + 5y \geq 15$
     Rewrite as $3x + 5y = 15$.

     • Set $x = 0$: $5y = 15 \Rightarrow y = 3$. Point: $(0, 3)$.
     • Set $y = 0$: $3x = 15 \Rightarrow x = 5$. Point: $(5, 0)$.

   • Constraint 2: $x + 3y \leq 9$
     Rewrite as $x + 3y = 9$.

     • Set $x = 0$: $3y = 9 \Rightarrow y = 3$. Point: $(0, 3)$.
     • Set $y = 0$: $x = 9$. Point: $(9, 0)$.

   • Check the origin (0,0) in both inequalities:

     • For $3x + 5y \geq 15$, we get $0 + 0 = 0 \geq 15$, which is false.
     • For $x + 3y \leq 9$, we get $0 + 0 = 0 \leq 9$, which is true.

   The feasible region is bounded by the points $(9, 0)$, $(5, 0)$, and $(0, 3)$.

2. Graph of the constraints:

3. Corner Points: The corner points of the feasible region are:
   $(9, 0)$, $(5, 0)$, and $(0, 3)$.

4. Evaluate the objective function at each corner point:

   • At $(5, 0)$:
     $z = 3(5) + 0 = 15$

   • At $(9, 0)$:
     $z = 3(9) + 0 = 27$

   • At $(0, 3)$:
     $z = 3(0) + 3 = 3$

5. Conclusion: The maximum value of $z$ is 27, which occurs at the point $(9, 0)$.

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Key Formulas or Methods Used

• Linear Programming: Maximize or minimize a linear objective function subject to linear constraints.
• Feasible Region: The region of the graph where all constraints are satisfied.
• Corner Point Theorem: The optimal value of a linear programming problem occurs at one of the corner points of the feasible region.

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Summary of Steps

1. Rewrite the constraints as equalities.
2. Solve for the intersection points of the constraints.
3. Plot the feasible region on the graph.
4. Identify the corner points of the feasible region.
5. Evaluate the objective function at each corner point.
6. Determine the maximum or minimum value of the objective function.

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