Linear Programming Formulation

Decision Variables:

• Let $x_1$: the quantity of product A to be produced (in kilograms).
• Let $x_2$: the quantity of product B to be produced (in kilograms).

Objective Function:

Minimize the total production cost: $\text{Minimize } Z = 2x_1 + 3x_2$

Constraints:

1. Total production constraint: $x_1 + x_2 \geq 350$
2. Minimum production of product A: $x_1 \geq 125$
3. Processing time constraint: $2x_1 + x_2 \leq 600$
4. Non-negativity constraints: $x_1 \geq 0, \, x_2 \geq 0$

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Solution Using Simplex Method

Step 1: Convert Constraints to Standard Form

Introduce slack and surplus variables to convert inequalities into equalities:

1. For $x_1 + x_2 \geq 350$, introduce a surplus variable $s_1$: $x_1 + x_2 - s_1 = 350, \quad s_1 \geq 0$
2. For $2x_1 + x_2 \leq 600$, introduce a slack variable $s_2$: $2x_1 + x_2 + s_2 = 600, \quad s_2 \geq 0$

Step 2: Initial Tableau

The initial simplex tableau is constructed with the variables $x_1, x_2, s_1, s_2$, and the objective function $Z$.

Let me calculate the optimal solution using the simplex method. scipy.optimize import linprog

Coefficients of the objective function (minimize cost: 2x1 + 3x2)

c = [2, 3]

Coefficients of the constraints (Ax <= b)

A = [ [-1, -1], # x1 + x2 >= 350 -> -x1 - x2 <= -350 [2, 1] # 2x1 + x2 <= 600 ]

b = [-350, 600]

Bounds for the variables

x_bounds = (125, None) # x1 >= 125 y_bounds = (0, None) # x2 >= 0

Solve the linear programming problem using the simplex method

result = linprog(c, A_ub=A, b_ub=b, bounds=[x_bounds, y_bounds], method='highs')

result### Solution

Optimal Product Mix

• $x_1 = 250$ kilograms (product A)
• $x_2 = 100$ kilograms (product B)

Total Minimum Cost

$Z = 2(250) + 3(100) = 800 \, \text{Rs.}$

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Does the Problem Have Multiple Optimal Solutions?

No, the problem does not have multiple optimal solutions. The simplex method identifies a unique optimal solution when the constraints and objective function intersect at a single vertex of the feasible region. Since there are no alternate optimal points along the objective function's line, this solution is unique.

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Would you like a detailed step-by-step explanation of the simplex method or further assistance with this problem?

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Further Questions to Explore:

1. How do the constraints affect the feasible region?
2. What happens if the cost of product B increases to Rs. 4/kg?
3. How sensitive is the solution to changes in processing time availability?
4. What if the minimum production of product A increases to 150 kg?
5. How can this solution be implemented practically in production planning?

Tip: Always verify feasibility of the solution by substituting values into the constraints.