Linear programming
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The document describes several linear programming problems involving constraints and objectives to maximize profit or minimize cost. The problems involve determining optimal production levels of products given constraints on available resources or determining optimal mixtures of ingredients given constraints.
Linear programming
- 1. Formulation of Linear Programming Model Identifying Decision Variables. Define Objective Function. Identifying Relevant Constraints.
- 2. A factory manufactures two articles “Chair” and “Table”. For chair a certain machine has to be worked for 3 hours and in addition a craftsman has to work for 4 hours. To manufacture the table, the machine has to be worked for 5 hours and in addition the craftsman has to work for 3 hours. In a week the factory can avail 160 hours of machine time and 140 hours of craftsman’s time. The profit on each chair is Rs.50 and that on each table is Rs.40. If all the articles produced can be sold, then find how many of chairs and tables be produced to earn the maximum profit per week. Formulate the problem as a Linear Programming Problem. Question
- 3. A company produces three products A, B and C from four raw materials P, Q, R and S. One unit of A requires 2 units of P, 3 units of Q and 4 units of S; one unit of B requires 4 units of Q, 3 units of R and 2 units of S; and one unit of C requires 4 units of P, 3 units of R and 4 units of S. The company has 12 units of material P, 14 units of material Q, 16 units of material R and 18 units of material S. Profit per unit of products A, B and C are Rs.4, Rs.7 and Rs.10 respectively. Formulate the problem as a Linear Programming Problem. Question
- 4. An electric appliance company produces two products: Refrigerators and ranges. Production takes place in two separate departments I and II. The company’s two products are sold on a weekly basis. The weekly production cannot exceed 25 refrigerators and 35 ranges. The company regularly employs a total of 60 workers in two departments. A refrigerator requires 2 man-weeks labour while a range requires 1 man-week labour. A refrigerator contributes a profit of Rs.60 and a range contributes a profit of Rs.40. Formulate the problem as a Linear Programming Problem. Question
- 5. Formulation of LPP model Objective function is, Maximize Z = 60x 1 + 40x 2 Subject to constraints, x 1 ≤ 25 x 2 ≤ 35 2x 1 + x 2 ≤ 60 x 1 ≥ 0 x 2 ≥ 0 Let x 1 = No. of units of Refrigerator produced. And let x 2 = No. of units of Range produced
- 6. Vitamins V and W are found in two different foods F 1 and F 2 . One unit of food F 1 contains 2 units of vitamin V and 3 units of vitamin W. One unit of food F 2 contains 4 units of vitamin V and 2 units of vitamin W. One unit of F 1 and F 2 cost Rs.3 and Rs.2.5 respectively. The minimum daily requirements (for one person) of vitamin V and W are 40 units and 50 units respectively. Assuming that anything in excess of daily minimum requirement of vitamin V and W is not harmful, find out the optimal mixture of food F 1 and F 2 at the minimum cost which meets the daily minimum requirements of vitamins V and W. Formulate this as a linear programming model. Question
- 7. Formulation of LPP model Objective function is, Minimize Z = 3x 1 + 2.5x 2 Subject to constraints, 2x 1 + 4x 2 ≥ 40 3x 1 + 2x 2 ≥ 50 x 1 ≥ 0 x 2 ≥ 0 Let x 1 = No. of units of food F1. And let x 2 = No. of units of food F2.
- 8. An animal feed company must produce exactly 200 kg of a mixture consisting of ingredients G1 and G2. The ingredient G1 costs Rs.3 per kg and G2 costs Rs.5 per kg. Not more than 80 kg of G1 can be used and atleast 60 kg of G2 must be used. Find the minimum cost mixture. Formulate this as a linear programming model. Question
- 9. Formulation of LPP model Objective function is, Minimize Z = 3x 1 + 5x 2 Subject to constraints, x 1 + x 2 = 200 x 1 ≤ 80 x 2 ≥ 60 x 1 ≥ 0 Let x 1 = No. of units (in Kg) of ingredient G1. And let x 2 = No. of units (in Kg) of ingredient G2.
- 10. Suppose you inherit Rs.100000 from your father-in-law that can be invested in a combination of only two portfolios, with the maximum investment allowed in either portfolio set at Rs.75000. The first portfolio has an average return of 10%, whereas the second has 20%. In terms of risk factors associated with these portfolios, the first has a risk rating of 4 (on a scale from 0 to 10), and the second has 9. Since you want to maximize your return, you will not accept an average rate of return below 12% or a risk factor above 6. Now, how much should you invest in each portfolio? Formulate this as a linear programming model. Question