The image provides details about a case study on a utensil manufacturer with a linear programming problem, followed by questions for discussion. The goal is to determine the optimal mix of utensil production given constraints on raw materials, machine time, and profitability. Let me break this down for you step by step.

1. Summary of the Problem

Objective Function:

Maximize $Z = 3x_1 + x_2 + 4x_3$

Where $x_1, x_2, x_3$ are the number of units of utensils Type I, II, and III to produce.

Subject to Constraints:

1. Raw Material Constraint:
   $6x_1 + 3x_2 + 5x_3 \leq 25$

2. Machine Time Constraint:
   $3x_1 + 4x_2 + 5x_3 \leq 20$

3. Non-Negativity Constraint:
   $x_1, x_2, x_3 \geq 0$

2. Questions for Discussion

The questions (a–d) involve exploring sensitivity analysis and variations in constraints and objective coefficients. Let’s analyze each:

(a) Impact of Profit Change for Type II Utensil

This explores how much the profit per unit of utensil Type II ($x_2$) can increase or decrease before affecting the optimal solution.

• The changes in profit values (≥1, ≥1.02, etc.) affect the current optimal basis used in the solution.

(b) Simplex Multiplier (Shadow Price) for Machine Time

The shadow price for the time constraint is $-3/5$.

• This indicates the rate of decrease in the objective function if the machine time availability is reduced by 1 unit (hour).
• The question explores the impact of increasing time availability within specified limits.

(c) Increase in Objective Function Due to Additional Machine Time

If machine time exceeds the given limit, calculate how the objective function $Z$ changes using the shadow price.

• Use the shadow price ($-3/5$) to calculate $Z$ changes when machine time increases to values like 25, 27.5, 35, etc.

(d) Profit Change for Type III Utensil

This explores the lower limit of the profit per unit of utensil Type III ($x_3$) while keeping the current basis optimal. Possible lower limits are:
4, 3, 2.5, or 2.

3. Approach to Solve

Each question requires a structured linear programming (LP) sensitivity analysis. Here's how to proceed:

For (a):

• Examine the sensitivity range for the profit coefficient of $x_2$.
• Compute how the optimal solution changes as $c_2$ (profit of Type II utensil) varies.

For (b):

• The shadow price is given as $-3/5$.
• Use this to assess how the constraint changes when machine time varies within the given range.

For (c):

• Compute the exact increase in $Z$ using the shadow price as machine time exceeds 20 units.

For (d):

• Check the lower limit of $c_3$ (profit of Type III utensil) that allows the current basis to remain optimal.

Would you like me to solve these questions step-by-step or focus on a specific one? Let me know!

Relative Questions

1. How does the profit of each utensil type affect the overall optimization problem?
2. What is the significance of shadow prices in linear programming?
3. How do changes in constraints (e.g., machine time) impact the feasibility of the solution?
4. What are the key assumptions in sensitivity analysis for linear programming?
5. Can you interpret the economic meaning of the constraints in the problem?

Tip

Shadow prices provide valuable insight into resource constraints—they show how much the objective function would improve or worsen by relaxing or tightening these constraints!