Self Test Questions: Linear Programming Graphical Method
Theory
1. Explain with the help of a suitable example, what do you understand by linear programming.
2. What are the characteristics and limitations of a linear programming problem?
3. What do you understand by graphical method? Give its limitations.
4. Fill in the blanks
- A linear programming problem has a well defined objective function which is ...... and which is to be ...... or .......
- The constraints in a linear programming problem arises due to limitation of ...... These are linear ...... or ......
- The solution of a linear programming problem indicates the right combination of ...... which ...... or ...... the objective function satisfying the various .......
Practical
Problem Formulation
1. Jumpin Ltd. has canned apple and bottled juice as its products with profit margin Rs. 2 and Rs. 1 respectively per unit. The following table indicates the labour, equipment and material to produce each product per unit.
| Bottled Juice | Canned Apple | Total | |
| Labour (man hours) | 3 | 2 | 12 |
| Equipment (machine hours) | 1 | 2.3 | 6.9 |
| Material (unit) | 1 | 1.4 | 4.9 |
Formulate the problem specifying the product mix, which will maximize profit without exceeding the various levels of resources.
2. The managing director of a small scale company decides to manufacture two products, P1 & P2, each of which is processed in two shops, viz. Machining shop (M) and Finishing shop (F). One unit of P1 takes 15 hours of machining and 24 hours of finishing shops. The corresponding requirements for P2 are 25 hours and 11 hours respectively in shops. The total available hours per day in M and F shops are 375 and 264 respectively. P1 gives a profit of Rs. 18 per unit and P2 Rs. 16 per unit. Formulate the above problem.
3. A company owns two flour mills, A and B, which have different production capacities for high, medium and flour. This company has entered a contract to supply flour to a firm every week with at least 12, 8 and 24 quintals of high, medium and low grade respectively. It costs the company Rs. 1000 and Rs. 800 per day to run mill A and B respectively. On a day, mill A produce 6, 2 and 4 quintals of high, medium and low grade flour respectively. Mill B produce 2, 2 and 12 quintals of high, medium and low grade flour respectively. How many days per week should each mill be operated in order to meet the contract order most economically.
4. Decibel Electronics produces two products A and B that are sold on a weekly basis. The weekly production cannot exceed 25 for product A and 35 for product B. The company employs a total of 80 workers. Product A requires 2 man-weeks of labour whereas B requires only 1. A gives a profit of Rs. 16 and B Rs. 40. Formulate the above LPP.
5. A company that produces soft drinks has a contract that requires that a minimum of 80 units of the chemical A and 60 units of the chemical B go into each bottle of the drink. The chemicals are available in a prepared mix from two different suppliers. Supplier X1 has a mix of 4 units of A and 2 units of B that costs Rs. 10, and supplier X2 has a mix of 1 units of A and 1 unit of B that costs Rs. 4. How many mixes from company X1 and company X1 should the company purchase to honour contract requirement and yet minimize cost?
Graphical Method
1. Maximize: z = 3x + 2y
subject to the constraints:
x - y ≤ 1
x + y ≥ 3
x, y ≥ 0
2. Maximize: z = x + y
subject to the constraints:
x + y ≤ 1
-3x + y ≥ 3
x, y ≥ 0
3. A manufacture of packing material, manufacturers two type of packing tins, round and flat. Major production facilities involved are cutting and joining. The cutting department can process 300 round tins or 500 flat tins per hour. The joining department can process 500 round tins or 300 flat tins per hour. If the contribution towards profit for around tin is the same as that of a flat tin what is that the optimum production level?
4. Maximize z = 3x1 - 4x2
subject to
x1 - x2 ≥
0
x2 ≤ 6
x1, x2 ≥ 0
5. Maximize z = 3x1 + 2x2
subject to
x1 + 2x2 ≤
2
2x1 + x2 ≥ 6
x1, x2 ≥ 0
6. Maximize z = 4x1 + 3x2
subject to
2x1 + 3x2 ≤
6
4x1 + 6x2 ≥ 24
x1, x2 ≥ 0
7. Maximize z = 3x1 + 2x2
subject to
2x1 - 3x2 ≥ 0
3x1 + 4x2 ≤ -12
x1, x2 ≥ 0
8. Maximize z = 50x1 + 60x2
subject to
2x1 + x2 ≤
300
3x1 + 4x2 ≤ 509
4x1 + 7x2 ≤ 812
x1, x2 ≥ 0
9. Minimize z = 2x1 + 1.7x2
subject to
0.15x1 + 0.10x2 ≥
1.0
0.75x1 + 1.70x2 ≥
7.5
1.30x1 + 1.10x2 ≥
10.0
x1, x2 ≥ 0
10. Maximize z = 3x1 + 2x2
subject to
2x1 - x2 ≥
2
x1 + 2x2 ≤ 8
x1, x2 ≥ 0
11. Minimize z = x + y
subject to
2x + y ≥ 12
5x + 8y ≥ 74
x + 6y ≥ 24
x, y ≥ 0
12. Minimize z = 2x1 + 3x2
subject to
-x1 + 2x2 ≤
4
x1 + x2 ≤ 6
x1 + 3x2 ≥ 9
x1, x2 ≥ 0
13. Minimize z = 3x + 4ysubject to
5x + 8y ≤ 2000
3x + 10y ≤ 1000
x, y ≥ 0
14. Minimize z = 7x1 + 3x2
subject to
x1 + 2x2 ≥
3
x1 + x2 ≤ 4
0 ≤ x1 ≤
15/2
0 ≤ x2 ≤
3/2
15. Minimize z = 4x1 + 3x2
subject to
2x1 + 3x2 ≤ 6
2x2 ≤ 5
-3x1 + 2x2 ≤ 3
2x1 + x2 ≤ 4
x1, x2 ≥ 0
