VI Sem ECE Assignment of Optimization Techniques

VI Sem ECEAssignment of Optimization TechniquesUNIT-1Topics Covered1. Historical Development2. Engineering Applications of Optimization3. Classification of optimization problem.4. Formulation of Optimization Problem.Q.1. What is Operations Research?Q.2. Write the historical background of Optimization?Q.3. What are the engineering applications of Optimization?Q.4. Explain the classification of optimization problem.Q.5. Explain the formulation of optimization problem.UNIT-2Topics covered1. Linear Programming2. Definitions3. Formulation of Problem4. Graphical solution5. Simplex Method6. Two-Phase Method7. Big-M Method(penalty method)8. Revised Simplex method9. Duality10. Special Cases (Alternative solution, unbounded solution, infeasible solution and degeneracy)

DefinitionsQ.1. Explain Linear Programming Problem. Define the following terms:a) Decision Variablesb) Objective Functionc) Constraintsd) Non-negativity Restrictionse) Basic Variablesf) Non-basic Variablesg) Basic Solutionh) Basic Feasible Solutioni) Optimal SolutionLinear Programming FormulationQ.2. A garment manufacturer has a production line making two styles of shirts. Style I requires 200grams of cotton thread, 300 grams of Dacron thread, and 300 gram of linen thread. Style II requires 200grams of cotton thread, 200 grams of Dacron thread and 100 grams of linen thread. The manufacturesmakes a net profit of Rs.19.50 on Style I, Rs.15.90 on Style II. He has in hand an inventory of 24 Kg ofcotton thread, 26 Kg of Dacron thread and 22 Kg of linen thread. His immediate problem is to determinea production schedule, given the current inventory to make a maximum profit. Formulate the LPPmodel.Q.3. A firm makes two types of furniture : chairs and tables. The contribution for each product ascalculated by the accounting department is Rs.20 per chair and Rs.30 per table. Both products areprocessed on three machines M1, M2, M3. The time required by each product and total time availableper week on each machine are as follows:Machine Chair Table Available HoursM1 3 3 36M2 5 2 50M3 2 6 60How should the manufacturer schedule his production in order to maximize contribution?Q.4.An animal feed company must produce 200 Kg of a mixture consisting of ingredients X1 and X2daily. X1 cost Rs.3 per Kg and X2 Rs.8 per Kg. Not more than 80 Kg of X1 can be used and atleast 60 Kg

of X2 must be used. Find how much of each ingredient should be used if the company wants to minimizethe cost.Q.5. A medical scientist claims to have found a cure for the common cold that consists of three drugscalled K, S and H. His results indicate that the minimum daily adult dosage for effective treatment is 10mg of drug K, 6mg of drug S and 8 mg of drug H. Two substances are readily available for preparing pillsor drugs. Each unit of substance A contains 6mg, 1mg and 2mg of drugs K, S, and H respectively andeach unit of substance B contains 2mg, 3mg and 2mg of the same drugs. Substance A costs Rs.3 per unitand substance B costs Rs.5 per unit.Find the least-cost combination of the two substances that will yield a pill designed to contain theminimum daily recommended adult dosage.Q.6.Solve graphically the following LP problemGraphical solutionMaximize Z = 30 x 1 + 20 x 2Subject to-x 1 - x 2 ≥ -8-6 x 1 -4 x 2 ≤ -125 x 1 +8 x 2 = 20x 1 , x 2 ≥ 0Q.7.Solve graphically the following LP problemMaximize Z = 50 x 1 + 60 x 2Subject to2x 1 + x 2 ≤ 3003 x 1 + 4x 2 ≤ 4804 x 1 +7 x 2 ≤ 812x 1 , x 2 ≥ 0(a)Infeasible Problems

Q.8. Solve graphically the following LP problemMaximize 4 x 1 + 3 x 2Subject to2x 1 +3 x 2 ≤ 64 x 1 + 6 x 2 ≥ 24x 1 , x 2 ≥ 0Q.9. Solve graphically the following LP problemMaximize 3 x 1 + 2 x 2Subject to2x 1 - 3 x 2 ≥ 03 x 1 + 4 x 2 ≤ -12x 1 , x 2 ≥ 0(b) Multiple Solution ProblemQ.10. Solve graphically the following LP problemMaximize x 1 + x 2Subject to-2x 1 + x 2 ≤ 1x 1 ≤ 2x 1 + x 2 ≤ 3x 1 , x 2 ≥ 0Q.11. Differentiate between surplus , slack variables and artificial variables?Q.12. What is degeneracy problem in relation to linear programming problem?

Simplex MethodQ.13. Solve the L.P. ProblemMaximize x 1 + 2 x 2 + x 3Subject to2x 1 + x 2 - x 3 ≤ 22 x 1 - x 2 + 5 x 3 ≤ 64x 1 + x 2 + x 3 ≤ 6x i ≥ 0, i=1,2,3Q.14. Solve the L.P. ProblemMaximize Z = 2 x 1 + 6 x 2Subject to-x 1 + x 2 ≤ 12 x 1 + x 2 ≤ 2x 1 , x 2 ≥ 0Q.15. Solve the L.P. ProblemMaximize Z = 6 x 1 + 8 x 2Subject to2x 1 + 3x 2 ≤ 164 x 1 + 2x 2 ≤ 16x 1 , x 2 ≥ 0Q.16. Solve the L.P. ProblemMinimize x 1 - 3 x 2 + 2x 3

Subject to3x 1 - x 2 + 3x 3 ≤ 7-2 x 1 + 4 x 2 ≤ 12-4x 1 + 3x 2 + 8x 3 ≤ 10x i ≥ 0, i=1,2,3Q.17. Use Big-M method to solve the L.P. ProblemBig-M MethodMaximize x 1 + 2 x 2 + 3 x 3 – x 4Subject tox 1 + 2 x 2 + 3 x 3 =152 x 1 + x 2 + 5 x 3 = 20x 1 + 2 x 2 + x 3 + x 4 =10x i ≥ 0, i=1,2,3,4Q.18. Use Big-M method to solve the L.P. ProblemMaximize 30x 1 + 20 x 2Subject to-x 1 - x 2 ≥ -8-6 x 1 -4 x 2 ≤ -125x 1 + 8 x 2 = 20x i ≥ 0, i=1,2Q.19. Use Big-M method to solve the L.P. ProblemMaximize -2x 1 - x 2Subject to3x 1 + x 2 = 3

4 x 1 + 3 x 2 ≥ 6x 1 + 2 x 2 ≤ 4x i ≥ 0, i=1,2Q.20. Solve by two phase method the L.P. ProblemTwo Phase MethodMinimize Z = 2 x 1 + 9 x 2 + x 3Subject tox 1 + 4x 2 + 2x 3 ≥ 53x 1 + x 2 + 2x 3 ≥ 4x i ≥ 0, i=1,2,3Q.21. Solve by two phase method the L.P. ProblemMinimize Z = 4 x 1 + x 2Subject to3x 1 + x 2 = 3x 1 + 2x 2 ≤ 44x 1 + 3x 2 ≥ 6x i ≥ 0, i=1,2Q.22. Solve by two phase method the L.P. ProblemMaximize Z = 5 x 1 + 8x 2Subject to3x 1 + x 2 ≥ 3x 1 + 4x 2 ≥ 4x 1 + x 2 ≤ 5x i ≥ 0, i=1,2

Revised Simplex MethodQ.23. Use revised simplex method to solve the following L.P. ProblemMaximize Z = 3 x 1 + 2 x 2 + 5x 3Subject tox 1 + 2x 2 + x 3 ≤ 4303x 1 + 2x 3 ≤ 460x 1 + 4x 2 ≤ 420x i ≥ 0, i=1,2,3Q.24. Use revised simplex method to solve the following L.P. ProblemMaximize Z = 6 x 1 - 2 x 2 + 3x 3Subject to2x 1 - x 2 + 2x 3 ≤ 2x 1 + 4x 3 ≤ 4x 1 + 4x 2 ≤ 420x i ≥ 0, i=1,2,3Q.25. Use revised simplex method to solve the following L.P. ProblemMaximize Z = x 1 + 2 x 2Subject tox 1 + x 2 ≤ 3x 1 + 2x 2 ≤ 53x 1 + x 2 ≤ 6x i ≥ 0, i=1,2

DualityQ.26. Write the dual of the following L.P. ProblemMinimize Z = 3 x 1 -2 x 2 + 4x 3Subject to3x 1 + 5x 2 + 4x 3 ≥ 76x 1 + x 2 +3x 3 ≥ 47x 1 - 2x 2 - x 3 ≤ 10x 1 - 2 x 2 +5x 3 ≥ 34x 1 + 7 x 2 - 2x 3 ≥ 2x i ≥ 0, i=1,2,3Q.27. Find the dual of the following L.P. ProblemMaximize Z = 3 x 1 - 2x 2Subject tox 1 ≤ 4x 2 ≤ 6x 1 + x 2 ≤ 5-x 1 ≤ -1x i ≥ 0, i=1,2Q.28. Use duality to solve the following L.P. ProblemMinimize Z = 10 x 1 + 6 x 2 + 2x 3Subject to

-x 1 + x 2 + x 3 ≥ 13x 1 + x 2 - x 3 ≥ 2x i ≥ 0, i=1,2,3Q.29. Solve the dual of the following problem by simplex methodMaximize Z = 2 x 1 + x 2Subject tox 1 + 4x 2 ≤ 24x 1 + 2x 2 ≤ 142x 1 - x 2 ≤ 8x 1 - x 2 ≤ 3x i ≥ 0, i=1,2Special Cases (Alternative solution, unbounded solution, infeasible solution and degeneracy)Q.30. Solve the following problem by simplex methodMaximize Z = 2 x 1 + x 2Subject to4x 1 + 3x 2 ≤ 124x 1 + x 2 ≤ 84x 1 - x 2 ≤ 8x i ≥ 0, i=1,2Q.31. Solve the following L.P. ProblemMaximize Z = x 1 + 2 x 2 + 3x 3 - x 4Subject tox 1 + 2x 2 + 3x 3 = 152x 1 + x 2 +5 x 3 = 20

x 1 + 2x 2 + x 3 + x 4 = 10x i ≥ 0, i=1,2,3,4Q.32. Solve the following problem by simplex methodMaximize Z = 2 x 1 + x 2Subject tox 1 - x 2 ≤ 102x 1 - x 2 ≤ 40x i ≥ 0, i=1,2Q.33. Solve the following problem by simplex methodMaximize Z = 3 x 1 + 2 x 2Subject to2x 1 + x 2 ≤ 23x 1 + 4x 2 ≥ 12x i ≥ 0, i=1,2