Linear Programming _set1

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______________________________________________________________________ L inear Programming ___

__ Compiled By : Dass Sir….for any queries..ca l l at 9662464238 or mail at [email protected] _____1

ASSIGNMENT NO 1

Solve the following Linear Programming Problems Graphically

Feasible solution : Any solution that satisfies the non – negative restrictions of thegeneral LPP , is called feasible sol.

Major steps in the sol. of LPP by Graphical Method.

1. State the problem mathematically.2. Plot a graph representing all the constraints of the problem and identify the feasible

region(solution space ).3. Determine the co-ordinates of all corner points of the feasible region.4. Find out the value of the objective function at each corner (solution) points determi-

ned in step 3.5. Select the corner point that optimizes (maximizes or minimizes)the values of the

objective function. It gives the optimum feasible solution.The above method is a lso know n as Search approach.

Case I : Maximizatizon case : {Feasible solution

}

1. Max: z = 3x + 4y

subject to the constraints :

x + y 450 , 2x + y 60 , x 0 , y 0

Ans : x = 0 , y = 450 and max z = 1800.

2. Max: z = 5x + 4y


1.5x + 2.5y 80 , 2x + 1.5y 70 , x 0 , y 0

Ans : x = 20, y = 20 and max z = 180

3. Max: z = 20x + 30y


3x + 3y 36 , 5x + 2y 50 , 2x + 6y 60 , x 0 , y 0

Ans : x = 3 , y = 9 and max z = 330

4. Max: z = 7x + 10y


x + y 30000 , y 12000 , x 6000 , x y , y 0 , x 0 Ans : x = 18000 , y = 12000 and max z = 246000.

5. Max: z = 40x + 80y


2x + 3y 48 , x 15 , y 10 , x 0 , y 0

Ans : x = 9 , y = 10 and max z = 1160.





continued assignment No : 1

6. Max: z = 40x + 50y


x + y 800 , 2x + y 1000 , x 400 , y 700 , x 0 , y 0 Ans : x = 100 , y = 700 and max z = 30000.

7. Max: z = 2x + 3y


3x + 4y 60 , y 3 , x 2y , x 0 , y 0

Ans : x = 12, y = 6 and max z = 42

8. Max: z = 10x + 15y


x + 2y 600 , x + 4y 800 , 3x + 20y 3600 x 500 , x 0 , y 0

Ans : x = 500, y = 50 and max z = 5750

9. Max: z = 3x + 2ysubject to the constraints :

-2x + y 1 , x 2 , x + y 3 , y 0 , x 0

Ans : x = 2 , y = 1 and max z = 8.

10. Max: z = 4x + 3y


2x + 3y 6 , - 3x + 2y 3 , 2x + y 4 , 2y 5 , y 0 , x 0

Ans : x = 1.5 , y = 1 and max z = 9

11. Max: z = 5x + 7y


x + y 4 , 3x + 8y 24 , 10x + 7y 35 , y 0 , x 0

Ans : x = 1.6 , y = 2.4 and max z = 24.8

12. Max: z = 2x + 3y


x + y 30 , x - y 0 , 0 x 20 , 3 y 12

Ans : x = 18 , y = 12 and max z = 72

13. Max: z = 50x + 30y


2x + y 18 , x + y 12 , 3x + 2y 34 , x 0 , y 0

Ans : x = 10 , y = 2 and max z = 560.





continued assignment No : 1

14. Max: z = 10x + 6y


3x + y 12 , 2x + 5y 34 , y 0 , x 0

Ans : x = 20 , y = 0 and max z = 200

15. Max: z = 90x + 60y


5x + 8y 2000 , 7x + 4y 1400 , y 0 , x 0

Ans : x = 800/9 , y = 1750/9 and max z = 19666.6





ASSIGNMENT NO 2

Solve the following Linear Programming Problems Graphically

Case II : Minimization case : { Feasible solution }

1. Min: z = 3x + 6y


- x + y 6 , x + y 10 , x 0 , y 0

Ans : x = 10 , y = 0 and min z = 30.

2. Min: z = 6x + 4y


- x + y 1 , x + y 3 , x 0 , y 0

Ans : x = 1 , y = 2 and min z = 14.

3. Min: z = 200x + 400ysubject to the constraints :

x + y 200 , x + 3y 400 , x + 2y 350 , x 0 , y 0

Ans : x = 100, y = 100 and min z = 6000

4. Min: z = x + y


2x + y 12 , 5x + 8y 74 , x + 6y 24 , x 0 , y 0

Ans : x = 2, y = 8 and min z = 10

5. Min: z = 600x + 400y


3x + 3y 40 , 3x + y 40 , 2x + 5y 44 , x 0 , y 0

Ans : x = 12, y = 4 and min z = 8800

6. Min: z = 20x + 40y


36x + 6y 108 , 3x + 12y 36, 20x + 10y 100 , x 0 , y 0

Ans : x = 4, y = 2 and min z = 160

7. Min: z = 0.60x + y


10x + 4y 20, 5x + 5y 20 , 2x + 6y 12 , x 0 , y 0

Ans : x = 3, y = 1 and min z = 2.80





ASSIGNMENT NO 3

SOME EXCEPTIONAL CASES :

A. Verify that that the following problems have no feasible solution.

1. Max: z = 15x + 20y


x + y 12 , 6x + 9y 54 , 15x + 10y 90 , x 0 , y 0

2. Max: z = 4x + 3y


x + y 3 , 2x - y 3 , x 4 , x 0 , y 0

B. Verify that that the following problems have multiple solution.

1. Max: z = 6x + 4y


x + y 5 , 3x + 2y 12 , x 0 , y 0

2. Max: z = 12x + 9y


3x + 6y 36 , 4x + 3y 24 , x + y 15 , x 0 , y 0

3. Max: z = 4x + 3y


3x + 4y 24 , 8x + 6y 48 , x 5 , y 6 , x 0 , y 0

C. Verify that that the following problem have an unbounded solution.

Max: z = 20x + 30y


5x + 2y 50 , 2x + 6y 20 , 4x + 3y 60 , x 0 , y 0

D. Graph the feasible region of the following problem and identify the redundant constraint.

Max: z = 5x + 6y


x + 2y 4 , 4x + 5y 20 , 3x + y 3 , x 0 , y 0





ASSIGNMENT NO 4

Formulate and solve the following Linear Programming Word Problems

Graphically :

1. A furniture dealer deals in two items : tables and chairs . He has Rs 5000 to invest and a space tostore at most 60 pieces . A table costs Rs 250 and a chair costs Rs 50 . He can sell a table at a

profit of Rs 50 and a chair at a profit of Rs 15. Assuming that he can sell all the items he buys,how should he invest his money so that he may maximize his profit ? Ans : He must purchase 10tables and 50 chairs.

2. Two tailors X and Y earn Rs 300 and Rs 400 per day resp. X can stitch 6 shirts and 4 pants whileY can stitch 10 shiirts and 4 pants per day. How much days does each of them work , if it desiredto produce at least 60 shirts and 32 pants at a minimum labour cost? Ans 5 days & 3 days resp.

3. A man has Rs 1500 for purchase of rice and wheat . a bag of rice and a bag of wheat costs Rs180 and Rs 120 resp. He has a storage capacity of 10 bags only . He earns a profit of Rs 11 andRs 8 per bag of rice and wheat respectively . How many bags of each must he buy to make max

profit ? Ans : 5 bags of each.

4. A manufacturer produces two products A and B and has has machines in operation 24 hrs a dayProduction of A requires 2 hours of processing on machine M1 and 6 hours on machine M2 .Production of B requires 6 hours of processing on machine M1 and 2 hours on machine M2. Themanufacturer earns a profit of Rs 5 on each unit of A and Rs 2 on each unit of B. How manyunits of each product should be produced in order to achieve maximum profit ?. Ans :3 units ofeach.

5. A company produces two types of presentation goods A and B that requires gold and silver .each unit of A requires 3 gms of silver and 1 gm of gold while that of A requires 1 gm of silver

and 2 gms of gold. The company can procure 9 gms of silver and 8 gms of gold . If each unit oftype A brings a profit of Rs 40 and that of types B Rs 50, determine the number of units of eachtype that the company should produce to maximize the profit. What is the maximum profit ?

Ans : 2 units of type A and 3 units of type B , max profit is Rs 230.

6 A dealer wishes to purchase a number of fans and sewing machines. He has Rs 5760 to investand his space utmost for 20 items. A fan costs him Rs 360 and a sewing machine Rs 240. Hisexpectation is that he can sell a fan at a profit of Rs 22. and a sewing machine at a profit of Rs18. Assuming that he can sell all the items that he can buy , how should he invest his money inorder to maximize his profit ? Ans : The dealer should buy 8 fans and 12 sewing machines.

7. A toy company manufactures two types of dolls : A and B . Each doll of B takes twice as long toproduce as one of type A and the company would have the time to make a maximum of 2000per day if it produced only the basic version. The supply of plastic is sufficient to produce 1500dolls per day (both A and B combined) . Doll B requires a fancy dress of which there are only600 per day available If the company makes a profit of Rs 15 and Rs 25 per doll , respectively ,on dolls A and B , how many of each should be produced per day in order to maximize the profit?

Ans : 900 dolls of type A and 600 dolls of type B , max profit Rs 28,500.





continued assignment No :4.

8. If a young man rides his motor cycle at 25 kmph , he has to spend Rs 2 per km on petrol ; if herides it at a faster speed speed of 40 kmph , the petrol cost increases to Rs 5 per km . He has Rs100 to spend on petrol and wishes to find the max distance he can run within one hr. Express thisas LPP and then solve it. Ans : The young man can cover the max distance of 30 km , if he rides50/3 km at 25 kmph and 40/3 km at 40 kmph.

9. Suppose every gram of wheat provides 0.1g of proteins and 0.25g of C6H12O6 and the corresp-onding values for rice are 0.05 g and 0.5g resp. Wheat costs Rs 5 and rice Rs 20 per kg. Theminimum daily requirements of proteins and C6H12O6 for an average man are 50g and 200gresp. In what quantities should wheat and rice be mixed in the daily diet to provide the minimumdaily requirements of proteins and C6H12O6 in minimum cost , assuming that both wheat and riceare to be taken in the diet ? Ans : 400g of wheat and 200g of rice.

10. A diet for a sick person must contain at least 4000 units of vitamins , 50 units of minerals and1400 calories. Two foods A and B are available at a cost of Rs 4 and Rs 3 per unit respectively.If one unit of A costs contains 200 units of vitamins, 1 unit of mineral and 40 calories and 1 unit ofood B contains 100 units of vitamins , 2 units of minerals and 40 calories, find by graphic

method, what combination of foods be used to have least cost. ? Ans : The diet should contain 5units of food A and 30 units of food B.

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Linear Programming _set1