Linear Programming Problems

A software company is working on two new IT projects – Project A (Mobile App) and Project B (Web Portal). Each project generates profit contributions of Rs. 20,000 per unit for Project A and Rs. 30,000 per unit for Project B. Both projects require resources from three specialized departments: Design (D1), Programming (D2), and Testing (D3). Project A requires 3 hours of design department, 5 hours of programming department and 2 hours of testing department while Project B requires 3 hours of design department, 2 hours of programming department and 6 hours of testing department. The available time in hours per week are 36, 50 and 60 for the department of design, programming and testing respectively. Formulate this problem as a L.P.P. How should the company schedule his production in order to maximize contribution? Use simplex method.[10]

Find the optimum solution of the given LPP by using the simplex method.Min Z=20A+10BMin; Z = 20A + 10BMinZ=20A+10BSubject to:Subject;to:Subjectto:A+2B≤40A + 2B \le 40A+2B≤404A+3B≥604A + 3B \ge 604A+3B≥603A+B≥303A + B \ge 303A+B≥30A,B≥0A, B \ge 0A,B≥0[10]

Solve the given linear programming problem (LPP) by using simplex method and interpret the findings.

Maximize Z = $4A+3B+6C$

Subject to Constraints:

$2A+3B+2C \leq 440$

$4A+3C \leq 470$

$2A+5B \leq 430$

$A,B,C \geq 0$

[10]

Solve the given Linear Programming Problem (LPP) by using simplex method and interpret the results.

Minimize $Z = 25A + 10B$

subject to constraints:

$$\begin{aligned} A + B &= 50 \ A &\geq 20 \ B &\leq 40 \end{aligned}$$

Where $A, B \leq 0$

[10]

Solve the given linear programming problem (LPP) by using simplex method and interpret the findings.

Min Z = $20A+10B$

Subject to

$A+2B \leq 40$

$4A+3B \geq 60$

$3A+B \geq 30$

$A,B \geq 0$

[10]

Solve the given Linear Programming Problem (LPP) by using simplex method and interpret the results.

Maximize: $Z = 10X_1 + 20X_2$

Subject to: $$4X_1 + 2X_2 \leq 60$$ $$4X_1 + 10X_2 \leq 100$$ $$2X_1 + 3X_2 \leq 38$$ $$X_1, X_2 \geq 0$$

[10]

The TechZone Software Company combines two key resources - Front-End Developers (A) and Back-End Developers (B) - to complete a software system that must involve exactly 150 person-hours of total work. Each Front-End Developer hour costs Rs. 2,000, and each Back-End Developer hour costs Rs. 8,000. The company must use at least 14 hours of Back-End work and no more than 20 hours of Front-End work in a project. Formulate objective function and constraints of this LPP. [5]

ABC Manufacturing Company produces three products: Tables, Chairs, and Desks. These products require processing through three departments: Cutting, Assembly, and Painting. These three departments have limited working time to 300 hours, 450 hours, and 200 hours per week respectively. Each table requires 4 hours for cutting, 5 hours for assembly, and 1 hour for painting and contributes Rs. 1000 to profit. Each chair requires 3 hours for cutting, 4 hours for assembly, and 1 hour for painting and contributes Rs. 800 to profit. Each desk requires 2 hours for cutting, 3 hours for assembly, and 2 hours for painting and contributes Rs. 1200 to profit. To maintain balance, the total production of all three products must not exceed 150 units per week. Formulate objective (profit) function and constraints for this LPP. [5]

Food X contains 6 units of vitamin A and 7 units of vitamin B and costs Rs. 5 per gram. Food Y contains 8 units of vitamin A and 12 units of vitamin B and costs Rs. 18 per gram. The daily minimum requirement of vitamins A and B are respectively 100 units and 138 units respectively. Formulate the problem as a LPP with the objective function minimizing the cost. [5]

Linear Programming Problem: Minimizing Cost of Food with Vitamin Requirements

Decision Variables: Let $x_1$, $x_2$, and $x_3$ be the units of foods $F_1$, $F_2$, and $F_3$ respectively.

Objective Function: Minimize $Z = 20x_1 + 24x_2 + 18x_3$

Constraints: $$20x_1 + 10x_2 + 10x_3 \geq 300 \quad \text{(Vitamin } V_1 \text{ requirement)}$$ $$10x_1 + 10x_2 + 10x_3 \geq 200 \quad \text{(Vitamin } V_2 \text{ requirement)}$$ $$10x_1 + 20x_2 + 10x_3 \geq 240 \quad \text{(Vitamin } V_3 \text{ requirement)}$$ $$x_1, x_2, x_3 \geq 0 \quad \text{(Non-negativity constraints)}$$

A food company at Kathmandu produce three types of a healthy food P, Q and R for children which contains three types of vitamin A, B, and C. Each unit of food P contains 2, 2 and 1 unit of vitamin A, B and C. One unit of food Q contains 2, 3 and 1 units while each unit of food R contains 1, 1 and 5 units of vitamin A, B and C respectively. Daily minimum requirements of vitamin A, B and C are 10, 12 and 14 units respectively. Formulate objective function and its constraints of LPP if cost per unit of food P, Q and R are Rs. 9, Rs. 12 and Rs. 15 respectively. [5]

The XYZ Company combines factors A and B to form a product which must weigh 50 pounds. At least 20 pounds of A and no more than 40 pounds of B can be used. The cost of A is Rs. 25 per pound and of B is Rs. 10 per pound. Formulate LPP to find the amount of factor A and B which should be used to minimize the cost. [5]