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  Development of LP Problems

Development of LP Problems

Example 1

A firm produces two kinds of products A and B and sells them at a gain of Rs. 2 on class A and Rs. 3 on class B. All products are processed on two machines G and H. Class A needs 1 minute of processing time on G and 2 minutes on H; Class B needs 1 minute on G and 1 minute on H. The machine G is obtainable for not more than 6 hours 40 minutes while machine H is obtainable for 10 hours for any working day. Prepare the problem as a linear programming problem.

 Answer

Assume

x1 be the number of products of class A

            x2 be the number of products of class B

 

After comprehending the problem, the known information can be methodically arranged in the form of the table given below.

 

Class of products (minutes)

 

Machine

Class A (x1 units)

Class B (x2 units)

Available time (mins)

G

1

1

400

H

2

1

600

Profit per unit

Rs. 2

Rs. 3

 

 

As the gain on class A is Rs. 2 per product, 2 x1 will be the gain on selling x1 units of class A. Likewise, 3x2 will be the gain on selling x2 units of class B. Thus, total gain (profit) on selling x1 units of A & x2 units of class B is given through

                                                Maximize Z = 2 x1+3 x2   (objective function)

 

As machine G takes 1 minute time on class A and 1 minute time on class B, the totality of minutes needed on machine G is given by   x1+ x2.

 

Likewise, the total number of minutes needed on machine H is given by 2x1 + 3x2.

 

But, machine G is obtainable for not more than 6 hours 40 minutes (400 minutes). Consequently,

                                                x1+ x2 ≤ 400                            first constraint 

 

In addition, the machine H is obtainable only for 10 hours that is 600 minutes, thus,

                                                2 x1 + 3x2 ≤ 600                       second constraint

 

As it is not feasible to manufacture negative quantities

                                                x1 ≥ 0 and x2 ≥ 0                      non-negative restrictions

 

 

 

Therefore

Maximize Z = 2 x1 + 3 x2

Subject to restrictions

                                    x1 + x2  ≤ 400

                                    2x1 + 3x2 ≤ 600

& non-negativity constraints              

x1 ≥ 0 , x2 ≥ 0

 Example 2

A firm manufactures two products A and B which have raw materials 400 quintals and 450 labour hours. It is identified that 1 unit of product A needs 5 quintals of raw materials and 10 man hours and gives a gain of Rs 45. Product B needs 20 quintals of raw materials, 15 man hours and gives a gain of Rs 80. Generate the LPP.

Answer

Assume

 x1 - number of units of product A

             x2 - number of units of product B

 

 

Product A

Product B

Availability

Raw materials

5

20

400

Man hours

10

15

450

Profit

Rs 45

Rs 80

 

 

Therefore

Maximize Z = 45x1 + 80x2

Subject to

5x1+ 20 x2 ≤ 400

                        10x1 + 15x2 ≤ 450

 x1 ≥ 0 , x2 ≥ 0

 

Example 3

A firm produces 3 products A, B and C. The profits are Rs. 3, Rs. 2 and Rs. 4 correspondingly. The firm has 2 machines and below is given the necessary processing time in minutes for each machine on each product.

 

 

Products

Machine

A

B

C

X

4

3

5

Y

2

2

4

Machine X and Y have 2000 and 2500 machine minutes. The firm must produce 100 A's, 200 B's and 50 C's type, but not more than 150 A's.

Answer

Assume

 x1 - number of units of product A

             x2 - number of units of product B

             x3 - number of units of product C

    

 

Products

 

Machine

A

B

C

Availability

X

4

3

5

2000

Y

2

2

4

2500

Profit

3

2

4

 

 

Max Z = 3x1 + 2x2 + 4x3

Subject to

4x1 + 3x2 + 5x3 ≤ 2000

                        2x1 + 2x2 + 4x3 ≤ 2500

                        100 ≤ x1 ≤ 150   

                        x2 ≥ 200

                        x3 ≥ 50

Example 4

A company possesses 2 oil mills A and B which have varied production capacities for low, high and medium grade oil. The company signs a contract to deliver oil to a firm every week with 12, 8, 24 barrels of each grade correspondingly. It costs the company Rs 1000 and Rs 800 per day to run the mills A and B. On a day A generates 6, 2, 4 barrels of each grade and B generates 2, 2, 12 barrels of each grade. Formulate an LPP to find out number of days per week each mill will be operated so as to meet the contract cost-effectively.

Answer          

Assume

x1 - no. of days a week the mill A has to work

x2 - no. of  days per week the mill B has to work

 

Grade

A

B

Minimum requirement

Low

6

2

12

High

2

2

8

Medium

4

12

24

Cost per day

Rs 1000

Rs 800

 

 

Minimize Z = 1000x1 + 800 x2

Subject to

 6x1 + 2x2 ≥ 12

                        2x1 + 2x2 ≥ 8

                        4x1 +12x2 ≥ 24

x1 ≥ 0 , x2 ≥ 0

Example 5

A company has 3 operational divisions processing, weaving and packing with the capacity to manufacture 3 different kinds of clothes that are suiting, shirting and woolen yielding with the gain of Rs. 2, Rs. 4 and Rs. 3 per meters correspondingly. 1m suiting needs 2 mins in processing 3mins in weaving and 1 min in packing. In the same way 1m of shirting needs 1 min in processing 4 mins in weaving and 3 mins in packing whereas 1m of woolen requires 3 mins in each division. In a week total run time of each department is 60, 40 and 80 hours for weaving, processing and packing department correspondingly. Develop a LPP to find the product to maximize the profit.

 

Answer

Assume

 x1 - number of units of suiting

 x2 - number of units of shirting

 x3 - number of units of woolen                      

 

 

Suiting

Shirting

Woolen

Available time

Weaving

3

4

3

60

Processing

2

1

3

40

Packing

1

3

3

80

Profit

2

4

3

 

Maximize Z = 2x1 + 4x2 + 3x3

Subject to

3x1 + 4x2 + 3x3 ≤ 60

                        2x1 + 1x2 + 3x3 ≤ 40

                        x1 + 3x2 + 3x3 ≤ 80

                        x1≥0, x2 ≥0, x3≥0

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