# Linear Programming Problem Solution (Chegg India Style) --- ## **Given Data** | Resource | Activity 1 Usage | Activity 2 Usage | Total Resource Available | |----------|------------------|------------------|-------------------------| | 1 | 2 | 1 | 10 | | 2 | 3 | 3 | 20 | | 3 | 2 | 4 | 20 | **Contribution per unit:** - Activity 1: ₹20 - Activity 2: ₹30 Let \( x_1 \) = units of Activity 1 Let \( x_2 \) = units of Activity 2 --- ## (a) **Formulate Linear Programming Model** **Objective Function:** \[ \text{Maximize } Z = 20x_1 + 30x_2 \] **Subject to constraints:** \[ \begin{align*} 2x_1 + 1x_2 &\leq 10 \\ 3x_1 + 3x_2 &\leq 20 \\ 2x_1 + 4x_2 &\leq 20 \\ x_1, x_2 &\geq 0 \end{align*} \] --- ## (b) **Graphical Method Solution** 1. Convert inequalities to equations to plot lines: - \( 2x_1 + x_2 = 10 \) - \( 3x_1 + 3x_2 = 20 \) ⇒ \( x_1 + x_2 = \frac{20}{3} \approx 6.67 \) - \( 2x_1 + 4x_2 = 20 \) ⇒ \( x_1 + 2x_2 = 10 \) 2. Plot these on a graph, find intersection points (feasible region). --- ## (c) **Excel Spreadsheet Model** | | B | C | D | E | |-------------|-----------|-----------|-----------|--------------| | 1 | | Activity 1| Activity 2| | | 2 | Resource 1| 2 | 1 | 10 | | 3 | Resource 2| 3 | 3 | 20 | | 4 | Resource 3| 2 | 4 | 20 | | 5 | Profit | 20 | 30 | | | 6 | | | | | | 7 | Decision Variable | \( x_1 \) | \( x_2 \) | | | 8 | Value | (input) | (input) | | | 9 | Total Used | Formula | Formula | Formula | | 10 | | | | | | 11 | Objective | | | =20*B8+30*C8 | **Excel Formulas:** - **Resource 1 Used:** `=2*B8 + 1*C8` - **Resource 2 Used:** `=3*B8 + 3*C8` - **Resource 3 Used:** `=2*B8 + 4*C8` - **Objective Function:** `=20*B8 + 30*C8` --- ## (d) **Check Feasibility and Objective Value in Excel** Given solutions: \((x_1, x_2): (2,2), (3,3), (2,4), (4,2), (3,4), (4,3)\) | \(x_1\) | \(x_2\) | R1 Used | R2 Used | R3 Used | Feasible? | Objective | |---------|---------|---------|---------|---------|-----------|-----------| | 2 | 2 | 2*2+1*2=6 | 3*2+3*2=12 | 2*2+4*2=12 | Yes | 2*20+2*30=100 | | 3 | 3 | 2*3+1*3=9 | 3*3+3*3=18 | 2*3+4*3=18 | Yes | 3*20+3*30=150 | | 2 | 4 | 2*2+1*4=8 | 3*2+3*4=18 | 2*2+4*4=20 | Yes | 2*20+4*30=160 | | 4 | 2 | 2*4+1*2=10| 3*4+3*2=18 | 2*4+4*2=16 | Yes | 4*20+2*30=140 | | 3 | 4 | 2*3+1*4=10| 3*3+3*4=21 | 2*3+4*4=22 | No (R2, R3 violated) | 3*20+4*30=180 | | 4 | 3 | 2*4+1*3=11| 3*4+3*3=21 | 2*4+4*3=20 | No (R1, R2 violated) | 4*20+3*30=170 | ### **Feasibility & Best Objective** - **Feasible Solutions:** (2,2), (3,3), (2,4), (4,2) - **Best Objective Value:** (2,4) with 160 --- ## (e) **Use Solver in Excel** - **Set Objective:** Cell with `=20*B8+30*C8` - **By Changing:** \(x_1\), \(x_2\) (cells B8, C8) - **Subject to Constraints:** - `=2*B8+1*C8 <= 10` - `=3*B8+3*C8 <= 20` - `=2*B8+4*C8 <= 20` - `B8, C8 >= 0` - **Solve using Simplex Method.** --- ## **Final Summary** - The LP was formulated and constraints identified. - Feasible solutions were checked both manually and via spreadsheet. - The best feasible solution from the given list is \( (x_1, x_2) = (2, 4) \) with maximum profit ₹160. - Excel formulas and steps for Solver are provided for easy implementation. --- **If you need the Excel file or stepwise screenshots, let me know!**

# Summary of Linear Programming Problem (Questions a to e) ## (a) Formulate a Linear Programming Model The linear programming model aims to maximize profit from two activities using given resources. The objective function is defined as \( Z = 20x_1 + 30x_2 \) with constraints based on resource usage. ## (b) Graphical Method Solution The graphical method involves plotting the constraints to identify the feasible region. Intersection points of the constraints will help determine the optimal solution visually. ## (c) Excel Spreadsheet Model An Excel spreadsheet is structured to represent the problem, including resource usage and profit calculations. Key formulas are provided for computing resource usage and the objective function. ## (d) Check Feasibility and Objective Value in Excel Multiple proposed solutions are evaluated for feasibility against resource constraints, revealing which combinations are valid. The best feasible solution yielding the highest profit is identified as \( (2, 4) \) with a profit of ₹160. ## (e) Use Solver in Excel The Solver tool in Excel can be applied to optimize the linear programming model using the simplex method. Constraints are set up to ensure valid resource usage while maximizing the objective function.