(a) Frame a Linear Programming Problem (LPP)

Let:

• $x_1$: Number of units of Product A
• $x_2$: Number of units of Product B

Objective Function: Maximize profit: $Z = 30x_1 + 40x_2$

Subject to constraints (based on raw material availability):

1. For raw material $P$: $2x_1 + 3x_2 \leq 600$
2. For raw material $Q$: $5x_1 + 3x_2 \leq 800$
3. For raw material $R$: $6x_1 + 5x_2 \leq 1100$

Non-negativity constraints: $x_1 \geq 0, \quad x_2 \geq 0$

(b) Solve Using Simplex Method

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

Coefficients of the objective function (profit maximization)

c = [-30, -40] # Negative sign because linprog performs minimization

Coefficients of the inequality constraints (LHS of inequalities)

A = [ [2, 3], # Raw material P constraint [5, 3], # Raw material Q constraint [6, 5] # Raw material R constraint ]

Right-hand side of the inequalities (availability of raw materials)

b = [600, 800, 1100]

Bounds for the variables (non-negativity constraints)

x_bounds = (0, None) y_bounds = (0, None)

Solve the problem using simplex method

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

Extract the results

optimal_profit = -result.fun # Negate because we maximized optimal_values = result.x

optimal_profit, optimal_values/tmp/ipykernel_12/1103920143.py:21: DeprecationWarning: method='simplex' is deprecated and will be removed in SciPy 1.11.0. Please use one of the HiGHS solvers (e.g. method='highs') in new code. result = linprog(c, A_ub=A, b_ub=b, bounds=[x_bounds, y_bounds], method="simplex") (8125.0, array([ 37.5, 175. ]))(8125.0, array([ 37.5, 175. ]))successThe optimal solution is as follows:

1. Optimal Profit: ₹8125
2. Optimal Production Plan:
   • Produce $x_1 = 37.5$ units of Product A.
   • Produce $x_2 = 175$ units of Product B.

Would you like a detailed explanation of the simplex steps or have further questions?

Related Questions:

1. What are the constraints in linear programming, and how are they applied here?
2. Why is the simplex method suitable for this problem?
3. How can we interpret the optimal solution graphically?
4. How do changes in raw material availability affect the optimal solution?
5. What if the profit per unit changes for one of the products?

Tip:

Always ensure your constraints align with real-world limitations to make practical use of optimization solutions!