Formulation

1. Production Allocation problem,A firm manufactures two types of products A& B and sell them at a profit of Rs 2 on type A and Rs 3 on type

B. Each product is processed on two machines G and H.Type A requires one minute of processing time on G And two minutes on H, type B requires one minute on G and one on H, the machine G is available for not more Than 6 hour 40 minutes while machine His available for 10 hours during any work day. Formulate the problem As LPP. 2. Dieticians tell us that a balance diet must contain quantities of nutrients such as calories, minerals. Vitamins The medical experts and dieticians tell that is necessary for an adult to consume 75g of proteins, 85g of fats, 300g of carbohydrates daily. The following table gives the food items, analysis, and their respective cost. Construct this problem as LPP to findout the food that should be recommended from a large number of Alternative sources of these nutrients so that the total cost of foot satisfying minimum requirements of Balanced diet is the lowest.3. A firm manufactures 3 products A, B and C. The profits are Rs 3,Rs2, &Rs4 respectively. The firm has two machines and below is given the required processing time in minutes for each machine on each product.

Machine

Products A B C

C D

4 3 5 2 2 4

Machines C and D have 2,000 and 2,500 machine-minutes respectively. The firm must manufacture 100 A’s, 200B’s and 50c’s but not more than 150 A’s. Formulate an LPP to maximize profit.

4) The manager of an oil refinery must decide on the optimum mix of 2 possible blending processes of which the inputs and production runs are as follows.

ProcessInput Out putCrude A Crude B Gasoline X Gasoline Y

1 2

6 5

4 6

6 5

9 5

The maximum amounts available of crudes A and B are 250 units and 200 units respectively. Market demand shows that at lest 150 units of gasoline X and 130units of gasoline Y must be produced. The profits per production run from process 1 and 2 are Rs 4 and Rs 5 respectively. Write the mathematical form of this problem for maximizing the profit.

5) A manufacturer produces 2 types of models M1 and M2 .Each M1 model requires 4 hrs of grinding and 2 hrs of polishing, where as each M2 Model requires 2hrs of grinding and 5hrs of polishing. The manufacturer has 2 grinders and 3 polishers. Each grinder works for 40 hrs per week and each polisher works for 60 hrs per week. Profit on M1 model is Rs 3, and on M2 model is Rs 4. What we produced in a week is sold in the market.How should the manufacturer should allocate his production capacity to the two types of models so that he mayMake the maximum profit. Write the mathematical form of the given problem.

6. An animal feed company must produce 200 Ibs of a mixture containing ingredients X1, X2,.X1 costs Rs 3 per Ib and X2 costs Rs 8 per Ib. Not more than 80 Ibs of X1 can be used and miximum quantity to be used for X2 is 60 Ibs. Formulate the mathematical form to find how much of each X1,X2 should be used if the company wants to minimize the cost.

7. Old hens can be bought for Rs 2 each but young hens cost Rs 5 each. The old hen lay 3 eggs per week each being worth 30 paise. A hen cost Rs 1 per week to feed. If have only Rs 80 to spend for hens, how many of eachKind should buy to give a profit of more than Rs 6 per week, assuming that I can’t house more than 20 hens.Write the mathematical form of the given problem.

8. A manufacturer produces two types of models M1 and M2. Each M1 model requires 4 hrs of grinding and 2 hrs of polishing where as each M2 model requires 2 hrs of grinding and 5 hrs of polishing. The manufacturer has 2

grinders and 3 polisher Each grinder works for 40 hrs per week and each polisher works for 60 hrs per week. Profit on M1 model is Rs 3 and on M2 model is Rs 4. What is produced in a week is sold in the market. HowShould the manufacturer should allocate his production capacity to the two types of models. So that he may make the maximum profit write the mathematical form of the given problem.

Graphical solution

1) Solve the given LPP Graphically Max Z = 3X1 + 4X2

S . to 4X1 + 2X2 80 2X1 + 5X2 180 X1 + X2 0.

2) Solve the given LPP Graphically.Min Z = 20X1 + 10X2 s.to

X1 + X2 40, 3X1 + X2 30 4X1 + 3X2 60 X1,X2 0.

3) Solve the given LPP Graphically. Max Z = 3X1 + 2X2 .

s.to X1 – X2 1 X1+ X2 3. X1 0, X2 0.

4) Solve the given LPP Graphically.. Min Z = 4X1 + 2X2

S.to X1 + 2X2 2 3X1 + X2 3 4X1 + 3X2 6 X1 X2 0

5) Solve the given LPP GraphicallyMax Z = X1 + X2 / 2

s.to 3X1 + 2X2 12 5X1 10 X1 + X2 8

- X1 + X2 4 X1, X2 0

6) Solve the LPP graphically Min is Z = X1 + X2 S.to X1 + X2 1

-3X1 + X2 3 X1, X2 0

7) Show graphically the maximum, minimum values of the objective function for the following are same.Z = 5X1 + 3X2 s.to X1 + X2 6 2X1 + 3X2 3 0 < X1 <3, 0 <X2 < 3.

8) Solve the following LPP’s graphically. Max Z = 2X2 – X1

s.to X1 – X2 -15X1 + X2 2 X2 2X1, X2

9) Max Z = 3X1 + 2X2

s.to – 2X1 + X2 1, X1 2, X1 + X2 3 X1, X2 0

10) Max Z = 45X1 + 80X2 s.to 5X1 + 20X2 400, 10X1 + 15X2 450 X1, X2 0

11) Max Z = 5X1 + 7X2 s.to X1 + X2 4, 3X1 +8X2 2, 10X1 + 7X2 = 35, X1, X2 0

12) Min Z = 3X1 + 5X2 s.to X1 4,2X2 6, 3X1 + 2X2 18, X1 + X2 9,X1, X2 0

Simplex

1. Using simplex method solve the following LPP. Max Z = X1 + 2X2 s.to

- X1 + 2X 2 8X1 + 2X2 12X1 – 2X2 3 X1, X2 0

2. Solve the given LPP using simplex method Max Z = 5X1 + 3X2 s.to X1 + X2 2 5X1 + 2X2 10 3X1 + 8X2 12

X1 , X2 0

3. Solve the LPP using simplex method.Min Z = X1 – X2 + X3 + X4 + X5 – X6

s.to X1 + X4 + 6X6 = 9 3X1 +X2 – 4X3 + 2X6 = 2 all Xi’s 0

4. Solve the given LPP using simplex method. Max Z = 107X1 + X2 + 2X3

s.to

14X1 +X2 –6X3 + 3X4 = 7 16X1 + 1/2X2 – 6X3 5 3X1 – X2 –X3 0 Xi’s 0

5. Using simplex method to solve the following LPP Max Z = 3X1 + 2X2 s.to X1 + X2 6, 2X1 + X2 6, X1,X2 0

6. Max Z = X1 + 2X2 + 3X3 S,to X1 2X2 + 3X3 10, X1 + X2 5, X1,X2 , X3 0.

7. Max Z = 5X1 + 2X2 + 3X3 – X4 + X5 s.to X1 + 2X2 +2X3 +X4 = 8 3X1 + 4X2 X3 + X5 = 7 X1 ,X2 ,X3 ,X4 ,X5 > 0

8. Max Z = -4X1,-3X2 –4X3 – 6X4 s.toX1 + 2X2 +2X3 + X4 80 2X1 + 2X3 + X4 60 3X1 +3X2 + X3 + X4 < 80

X1,X2,X3,X4, 0

9. Use Big-M method to solve the LPP.Min Z = 4X1 +3X2 s.to 2X1 + X2 10 -3X1 + 2X2 6 X1 + X2 6

X1,X2, 0

10.. Max Z = 5X1-4X2+3X3 s.to2X1 + X2 - 6X3 = 20 6X1 + 5X2 +10 X3 76 8X1 -3X2 +6 X3 50

X1,X2,X3,X4, 0

11. Solve the following LP Problems by two- phase method. Max Z = 3X1 – X2 s.to 2X1 + X2 2 X1 + 3X2 2 X2 4 & X1,X2 0

12. Min Z = X1- 2X2 – 3X3 s.to - 2X1 + X2 + 3X3 = 2 2X1 + 3X2 + 4X3 =1 Xi 0 I = 1,2,3.

13. Max Z = 5X1 + 8X2

s.to 3X1 +2X2 3X1 + 4X2 4X1 + X2 5X1 , X2 0

14. Max Z = 500X1 + 1400X2 + 900X3 s.to

X1 + X2 + X3 = 100 12X1+ 35X2 + 15X3 25

8X1 + 3X2 + 4X3 6 X1,X2 ,X3 0

Obtain the dual of the following linear programming problems.1. Max Z = 3X1+4X2

s.to 2X1+6X2 165X1+2X+2X2 20 X1, X2 0

2. Min Z= 7X1+3X2 +8X3

s.to 8X1+2X2 +X1 33X1+6X2+4X3 4

4X1+X2+5X31 X1+5X2+2X37 X1,X2 ,X3 0

3. . Max Z= 3X1+X2 +X3-X4

s.to X1+5X2 +3X3 +4X4 5X1+X2= -1

X3-X4-5 X1,X2,X3,X40

4. . Min Z= X3+X4 +X5

s.to X1-X3 +X4+X5 = -2 X2-X3-X4+X5= 1 Xi.>0,I=1,2…5

5. . Max Z= 6X1+4X2 +6X3+X4

s.to 4X1+4X2 +4X3 +8X4=213X1+17X2+80X3+2X448

X1,X2 ,0 ,X3 ,X4are unrestricted in sign.

6. Max Z = 3X1+2X2

s.to 2X1+X2 5 X1+X2 3 X1+X20.

7. . Max Z =2X1+X2

s.to X1+2X210 X1+X26 X1-X22.

X1-2X21X1,X2,0.

8. Min Z =2X1+2X2

s.to 2X1+4X21 X1+2X21 2X1+X21.

X1,X2, 0.

9. Solve the following Assignment problem .A department head four subordinates and four tasks have to be performed. Subordinates differ in efficiency and tasks differ in their intrinsic difficulty. Time each man would take to per form each task is given in the effectiveness matrix. How the tasks should be allocated to each person so as to minimize the total man.

Subordinates 1 11 11 IV A B Tasks c

10. A car hire company has one car at each of five depots a ,b,c,d and e.A customer requires a car in each town, namely A,B,C,D,E.Distance between depots and towns are given in the following distance matix.

A b c d e A B C D E

How should cars be assigned to customer so as to minimize the distance traveled?

11. Solve the assignment problem represented by the following matrix

a b c d e f

12. Solve the following assignment problem Man

a) 1 2 3 4 work

I II III IVA 10 12 9 11B 5 10 7 8C 12 14 13 11D 8 15 11 9

13. Solve the following cost minimizing problems.

I II III IV A 45 30 65 40 55B 50 30 25 60 30C 25 20 15 20 40D 35 25 30 30 20E 80 60 60 70 50

14. Solve the following Assignment problems.

8 26 17 1113 28 4 2638 19 18 1519 26 24 10

160 130 175 190 200135 120 130 160 175140 110 155 170 18550 50 80 80 11055 35 70 80 105

9 22 58 11 19 2743 78 72 50 63 4841 28 91 37 45 3374 42 27 49 39 3236 11 57 22 25 183 56 53 31 17 28

I 12 30 21 15II 18 33 9 31III 44 25 24 21IV 23 30 28 14

a)

W X Y Z AJob B

A B C D E F1 13 13 16 23 19 92 11 19 26 16 17 183 12 11 4 9 6 104 7 15 9 14 14 135 9 13 12 8 14 11

Jobs c)

Machines

d) Solve the following maximization assignment problem.

A B C D E1 32 38 40 28 402 40 24 28 21 363 41 20 33 30 374 22 38 41 36 365 29 33 40 35 39

A B C D EA 4 7 3 4B 4 6 3 4C 7 6 7 5D 3 3 7 7E 4 4 5 7

18 24 28 328 13 17 19

J10 15 19 22

A B C D E1 5 11 10 12 42 2 4 9 3 53 3 12 5 14 64 6 14 4 11 75 7 9 8 12 5

1 2 3 4A 32 41 57 18B 48 54 62 34C 20 31 81 57D 71 43 41 47E 52 29 51 50

15. Solve the following Travelling salesman peoplem.

e)A B C D E

A 2 5 7 1B 6 3 8 2C 8 7 4 7D 12 4 6 5E 1 3 2 8

b)A B C D E

A 2 4 7 1B 5 2 8 2C 7 6 4 6D 10 3 5 4E 1 2 2 8

c)1 2 3 4 5

1 6 12 6 42 6 10 5 43 8 7 11 34 5 4 11 55 5 2 7 8

A B C D E

d)A B C D E

1 32 38 40 28 402 40 24 28 21 363 41 20 33 30 374 22 38 41 36 365 29 33 40 35 39

ee

A B C D EA 4 7 3 4B 4 6 3 4C 7 6 7 5D 3 3 7 7E 4 4 5 7

16.Find the initial basic feasible solution of following transportation problem.Ware house W1 W2 W3 W4 Factory capacity

Factory 19 30 50 10 770 30 40 60 940 8 70 20 18

Ware house 5 8 7 14 34Requirement

By 1) North west corner rule2) The Row minima method3) The column minima method4) Matrix Minima method5) Vogel’s Approximation method

17) Determine an initial basic feasible solution to the following TP using north west corner rule.Avaliable

5 9 12 9 8 4 38 7 3 6 8 9 46 4 5 6 8 10 147 7 3 5 7 10 93 2 3 8 10 2 4

Require 3 4 5 7 6 4ment

18) Determine an initial basic feasible solution to the TP by row/Column minima method.

Available22 6 3 5 415 5 9 2 78 5 7 8 6

Demand 7 12 17 9

19) Obtain an initial basic feasible solution to the following TP using matrix minima method.

20) Determine an IBFS by voge’s methodSupply

4 2 11 10 3 78 1 4 7 2 19 3 9 4 8 12

Demand 3 3 4 5 6

21)

GodownsFactory 1 2 3 4 5 6 STOCK

availableA 7 5 7 7 5 3 60B 9 11 6 11 5 20C 11 10 6 2 2 8 90D 9 10 9 6 9 12 50

Demand 60 20 40 20 40 40Find I) The Initial Basic feasible solution by Vogel’s method ii) optimal basic feasible solution by modi’s method.

22.)Obtain an optimum BFS for the following TP

Supply

Capacity6 1 2 3 48 4 3 2 0

10 0 2 2 1Demand 4 6 8 6

7 7 10 5 11 454 3 8 6 13 909 8 6 7 5 95

12 13 10 6 3 755 4 5 6 12 105

Demand 20 80 50 75 85

23) Consider four basis of operations Bi and three Targets Tj. The ton of Bombs per aircraft from any base that can br delivered to any target are given in the following Table.

T1 T2 T3 AvailabilityB1 8 6 5 150B2 6 6 6 150B3 10 8 4 150B4 8 6 4 150

Requirment 200 200 200 200Find the allocation from each base to each target, which maximizes the total over all the three targets.