Academic Year 2008-09 (Semestter II)
Class: B. Tech. (Civil & WM Engg.) End Term Examination
Subject: Water Resourses Systems Engineering (WM 408)
Date: Dec. 2, 2008 / Time: 14.00 to 17.00 hrs / Max. Marks : 70
Notes: 1. Support your answers with neat sketches.
Q. 1. Solveany five of the following:(30)
(i)Explain the concept of system, its characteristics and hierarchy of systems with the help of an example from water resources.
(ii) What is systems approach? How does it differ from systems analysis? Explain in brief the steps in application of systems approach.
(iii) Explain in details the role of slack, surplus and artificial variables in solving linear programming problems using Simplex algorithm.
(iv) Classify the types of non-linear programming problems and explain the method of solution by Lagrange Multipliers.
(v) Discuss the characteristics and manageable resources of ground water system.How can the conjunctive use of surface and ground water be effectively achieved applying system concepts?
(vi) Explain in brief the theory of duality. What are primal-dual relationships? Discussthe concept of sensitivity analysis and its use.
Q. 2. (i) Solve the following LP problem by graphical method.(07)
Maximize Z = 4X1 + 5 X2
Subject to
X1≥1.5
2X2≥ 4.0
3X1 + 2X2≤ 18
What shall be the change in solution if the nature of objective function is minimization instead of maximization?
(ii) Solve the LP problem given in Q.2.(i) using Simplex method and check the answer. (13)
Q.3.(i)Compare Linear and Dynamic programming approaches in details.(07)
(ii) A total sum of three Rs. lakhs is to be allocated amongst three water resources proposals for augmenting irrigation water supply. The option of not allocating any amount to any proposal is open and the money is to be distributed in units of one lakh. The returns from the three proposals from investment is given in the following table. Find the optimum allocation and the maximum returns using Dynamic Programming. (13)
Investment / ProposalsA / B / C
0 / 0 / 1 / 0
1 / 1 / 1 / 1
2 / 1 / 3 / 3
3 / 3 / 4 / 3
OR
Q.3.(ii) Solve the following network problem to determine the shortest path between start and endusing Dynamic Programming. (13)
/ SGGS Institute of Engg. and Tech., NandedAcademic Year 2009-10 (Semestter II)
Class: M. Tech. (Civil) (WM Engg.) Mid Term Examination
Subject: Systems Engineering & its applications (MCW512)
Date: 24/02/2010 / Time: 13.00 to 14.30 hrs / Max. Marks : 30
Notes: 1. Support your answers with neat sketches.
Q. 1. Solve any three of the following:(12)
(i) Explain the concept of system and discuss its characteristics.
(ii) State various forms (equations) of a LP model and explain the terms.
(iii) Explain in details the role of slack, surplus and artificial variables in solving linear programming problems using Simplex algorithm.
(iv) Classify the types of non-linear programming problems and explain the method of solution by Hessian Matrix method.
(v) Discuss the concavity and convexity of functions.
Q. 2. (i) Solve the following LP problem (10)
Maximize Z = X1 + 2 X2+ X3
Subject to
2X1 + X2 - X3 ≤ 2
2X1 - X2 + 5X3 ≤6
4X1 + X2 + X3 ≤ 6
(ii) Find the extremities of the function(08)
Z = X13 + X23 + 2 X12 + 4 X22 + 6
Academic Year 2009-10 (Semester II)
Class: M. Tech. (Civil & WM Engg.) Summer Term Examination
Subject: Systems Engineering& its applications (MCW 512)
Date: 23rd June, 2010 / Time: 10.00 to 13.00 hrs / Max. Marks : 100
Q. 1. Solve any SIX of the following:(36)
(i) Define system and state its characteristics.
(ii) Explain the Bellman’s principle of optimality and state the requirements of DP problem?
(iii) What are various forms of LP model? State equations and explain the terms.
(iv) What is canonical form? How do you obtain it? Illustrate.
(v) Explain method of Lagrange Multipliers and compare it with method of direct substitution.
(vi) Explain Curse of Dimensionality for dynamic programming.
(vii) Discuss the concept of sensitivity analysis and its use.
Q. 2. Solve the following LP problem.(16)
Maximize Z = 2X1 + X2
Subject to3X1 + X2≤ 300
4X1 + 2X2≤ 500
Also write it's dual and solve. State the related complementary slack condition.
Q. 3. (a) Compare Linear and Dynamic programming approaches in details.(06)
(b) Solve the following LP problem by graphical method.(10)
Maximize Z = 4X1 + 5 X2
Subject toX1≥ 1.5
2X2≥ 4.0
3X1 + 2X2≤ 18
What shall be the change in solution if the nature of objective function is minimization instead of maximization?
Q.4.(a)Find the extremities of the function(06)
Z = 5X12 + 2X2 – X1X2
Subject to X1 + X2 = 3
(b) Discuss the concavity and convexity of a function.(06)
(c)State the composition of Hessian Matrix. How do you decide its nature?(04)
Q.5(a)Describe the transportation model and discuss its characteristics and applications.
State the methods for obtaining initial feasible solution and optimal solution.(10)
(b) Total available supply at each of the four Origins namely A, B, C and D is 10, 25, 25 and 30 respectively. The total demand at each of the four destinations is given as 20, 20, 15 and 35 respectively. Write the transportation array. Obtain initial feasible solution using transportation array and North West Corner Rule. (06)
/ SGGS Institute of Engg. and Tech., NandedAcademic Year 2009-10 (Semester II)
Class: M. Tech. (Civil & WM Engg.) End Term Examination
Subject: Systems Engineering& its applications (MCW 512)
Date: Apr., 24th 2010 / Time: 10.00 to 13.00 hrs / Max. Marks : 70
Notes: 1. Q.1 is compulsory. Solve any four questions from the remaining.
Q. 1. Solve any five of the following:(30)
(i) Explain the Bellman’s principle of optimality. What are the requirements of DP problem?
(ii) What is systems approach? How does it differ from systems analysis? Explain in brief the steps in application of systems approach.
(iii) What is canonical form? How do you obtain it? Illustrate.
(iv) Explain the method of solution by Lagrange Multipliers and compare it with method of direct substitution.
(v) Explain Curse of Dimensionality for dynamic programming
(vi) Discuss the concept of sensitivity analysis and its use.
(vii) Discuss the Kuhn-Tucker conditions for Non-linear programming problem.
Q. 2. Solve the following LP problem(10)
Maximize Z = 2X1 + X2
Subject to
3X1 + X2≤ 300
4X1 + 2X2≤ 500
Q. 3.(a) State primal-dual relationships and complementary slack condition.(6)
(b) Write the Dual for (4)
Maximize Z = 4X1 + 6 X2
Subject to
2X1 +4 X2≤ 15
X1 + 3X2≤ 20
X1≤ 10
Q.4.A total sum of Rs. three Lakhs is to be allocated amongst three water resources proposals for augmenting irrigation water supply. The option of not allocating any amount to any proposal is open and the money is to be distributed in units of one Lakh. The returns from the three proposals from investment are given in the following table. Find the optimum allocation and the maximum returns using Dynamic Programming. (10)
Investment(Rs. Lakh) / Returns from Proposals
Proposal A / Proposal B / Proposal C
0 / 0 / 0 / 0
1 / 1 / 1 / 1
2 / 1 / 3 / 3
3 / 3 / 4 / 3
Q.5. Find the extremities of the function(10) Z = (X1 – 4)2 + (X2 – 3)2
Subject to
36((X1 – 2)2 + (X2 – 3)2 = 9
Q.6 (a)State transportation model and discuss its characteristics. (05)
(b) Total available supply at each of the three Origins namely A, B and C is 22, 15 and 8 respectively. The total demand at each of the four destinations is given as 7, 12, 17 and 9 respectively. Obtain initial feasible solution using transportation array and North West Corner Rule. (05)
/ SGGS Institute of Engg. and Tech., NandedAcademic Year 2009-10 (Semestter II)
Class: M. Tech. (Civil) (WM Engg.) End Term Examination
Subject: Systems Engineering & its applications (MCW512)
Date: 28/04/2010 / Time: 14.00 to 15.30 hrs / Max. Marks : 70
Notes: 1. Support your answers with neat sketches.
Q. 1. Solve any three of the following:(12)
(i) Explain the concept of system and discuss its characteristics.
(ii) State various forms (equations) of a LP model and explain the terms.
(iii) Explain in details the role of slack, surplus and artificial variables in solving linear programming problems using Simplex algorithm.
(iv) Classify the types of non-linear programming problems and explain the method of solution by Hessian Matrix method.
(v) Discuss the concavity and convexity of functions.
Q. 2. (i) Solve the following LP problem (10)
Maximize Z = X1 + 2 X2 + X3
Subject to
2X1 + X2 - X3 ≤ 2
2X1 - X2 + 5X3 ≤ 6
4X1 + X2 + X3 ≤ 6
(ii) Find the extremities of the function(08)
Z = X13 + X23 + 2 X12 + 4 X22 + 6
/ SGGS Institute of Engg. and Tech., NandedAcademic Year 2010-11 (Semester II)
Class: M. Tech. (Civil & WM Engg.) (Old/Revised)
End Term Examination
Subject: Systems Engineering& its applications (MCW 513)
Date: Apr., 13th 2011 / Time: 10.00 to 13.00 hrs / Max. Marks : 70
Notes: 1. Q.1 is compulsory. Solve any four questions from the remaining.
Q. 1. Solve any five of the following:(30)
(i) What do you understand by Multistage Decision process? Give illustration for a civil engineering system. Explain the Bellman’s principle of optimality.
(ii)Compare systems analysis and systems approach? Explain steps in systems approach.
(iii) How do you convert given LP into its standard form? Also explain the canonical form.
(iv) Explain the Hessian Matrix method of solution to NLP problem. State the limitations.
(v) Explain Stage variable, state variable and recursive equation for a DP formulation.
(vi) Discuss the concept of sensitivity analysis and its use in practice.
(vii) Discuss the Kuhn-Tucker conditions for Non-linear programming problem.
Q. 2. Solve the following LP problem : Maximize Z = X1 + 2X2 + X3(10)
Subject to: 2X1 + X2-X3 ≤ 2
-2X1 + X2- 5X3 ≤ - 6
4X1 + X2+X3 ≤ -6and Nonnegativity Constraint
Q. 3. (a) State primal-dual relationships and complementary slack condition.(6)
(b) Write the Dual for Maximize Z = 3X1 + 5X2(4)
Subject to :2X1 +6X2≤ 50
3X1 + 2X2≤ 35
5X1 - 3X2≤ 10
X2≤ 20
Q.4. A total sum of Rs. three Lakhs is to be allocated amongst three water resources proposals for augmenting irrigation water supply. The option of not allocating any amount to any proposal is open and the money is to be distributed in units of one Lakh. The returns from the three proposals from investment are given in the following table. Find the optimum allocation and the maximum returns using Dynamic Programming. (10)
Investment(Rs. Lakh) / Returns from Proposals
Proposal A / Proposal B / Proposal C
0 / 0 / 0 / 0
1 / 1 / 1 / 1
2 / 1 / 3 / 3
3 / 3 / 4 / 3
Q.5. Find the extremities of the function and discuss the method used(10) Z = (X1 – 4)2 + (X2 – 3)2
Subject to
36 ((X1 – 2)2 + (X2 – 3)2 = 9
Q.6 (a)Explain Transportation array and discuss its noteworthy features.(05)
(b) Total available supply at each of the three Origins namely A, B and C is 10, 25 and 25 respectively. The total demand at each of the four destinations is given as 10, 15, 20 and 15 respectively. Obtain initial feasible solution using North West Corner Rule. (05)
/ SGGS Institute of Engg. and Tech., NandedAcademic Year 2011-12(Semester II)
Class: M. Tech. (Civil & WM Engg.) (Old/Revised)
End Term Examination
Subject: Systems Engineering& its applications (MCW 513)
Date: 02/05/2012 / Time: 14.00 to 17.00 hrs / Max. Marks : 70
Notes: 1. Q.1 is compulsory. Solve any four questions from the remaining.
Q. 1. Solve any five of the following:(30)
(i) Explain the Bellman’s principle of optimality. Illustrate requirements of DP.
(ii)Explain the method of direct substitution for NLP and discuss its limitations.
(iii) State the various forms of LP formulations and explain the terms. Discuss various transformations for converting given LP formulation into its standard form.
(iv) Explain the Hessian Matrix method of solution to NLP problem. State the limitations.
(v) Explain Stage variable, state variable and recursive equation for a DP formulation.
(vi) Discuss the concept of sensitivity analysis and its use in practice.
(vii) Discuss the Kuhn-Tucker conditions for Non-linear programming problem.
Q. 2. Solve the following LP problem : Maximize Z = 3X1 + 2X2(10)
Subject to: X1 + X2 ≤ 4
X1 - X2 ≤ 2
and Nonnegativity Constraint
Q. 3.State primal-dual relationships. Write the Dual and solve the following LP
Maximize Z = X1 + 2X2(10)
Subject to :2X1+ X2≥ 4
X1 + 7X2≥ 7
Q.4. A total sum of Rs. three Lakhs is to be allocated amongst three water resources proposals for augmenting irrigation water supply. The option of not allocating any amount to any proposal is open and the money is to be distributed in units of one Lakh. The returns from the three proposals from investment are given in the following table. Find the optimum allocation and the maximum returns using Dynamic Programming. (10)
Investment(Rs. Lakh) / Returns from Proposals
Proposal A / Proposal B / Proposal C
0 / 0 / 0 / 0
1 / 1 / 1 / 1
2 / 1 / 3 / 3
3 / 3 / 4 / 3
Q.5. Find the extremities of the function and discuss the method used(10) Z = 2X1 + X2 + 10
Subject to
X1 +2X22 = 3
Q.6 (a)Explain Transportation array and discuss its noteworthy features.(05)
(b) Total available supply at each of the three Origins namely A, B and C is 20, 35 and 15 respectively. The total demand at each of the four destinations is given as 15, 15, 25 and 15 respectively. Obtain initial feasible solution using North West Corner Rule. (05)
/ SGGS Institute of Engg. and Tech., NandedAcademic Year 2011-12(Semester II)
Class: M. Tech. (Civil & WM Engg.) (Old/Revised)
Summer Term Examination
Subject: Systems Engineering & its applications (MCW 513)
Date: 26 /06/2012 / Time: 14.00 to 17.00 hrs / Max. Marks : 100
Notes: 1. Q.1 is compulsory. Solve any four questions from the remaining.
Q. 1. Solve any Four of the following:(40)
(i) Explain requirements of DP formulations. Discuss Stage, State, and Recursive equation.
(ii) Explain the method of Lagrange Multiplier for NLP and discuss its limitations.
(iii) State the various forms of LP formulations and explain the terms. Explain briefly the simplex method and state the limitations.
(iv) Explain the Hessian Matrix. How do you decide its nature? Illustrate.
(v) Discuss the concept of sensitivity analysis and its use in practice.
(vi) Discuss the Kuhn-Tucker conditions for Non-linear programming problem.
Q. 2. Explain the Graphical method of solution. Solve and discuss the type of solution
for the following LP problem :
Maximize Z = 3X1 + 2X2(15)
Subject to: 3X1- 2X2 ≤ 6
X1 - X2 ≤ 1 and Nonnegativity Constraint
Q. 3.(a) State primal-dual relationships. Write the Dual the following LP
Minimize Z = 2X1 + X2(11)
Subject to : 3X1 + X2≥ 3
X1 + 2X2≥ 3
4X1 + 3X2≥ 6
(b) What is meant by complementary slack condition?(04)
Q.4.(a) A total available water of threeunits is to be allocated amongst three users for irrigation. The option of not allocating any amount to any user is open and the water is to be distributed in units of one unit or integer multiple. The returns from the user are given in the following table. Find the optimum allocation and the maximum total returns using Dynamic Programming. (09)
Water allocation / Returns from allocation to UserUser 1 / User 2 / User3
0 / 0 / 0 / 0
1 / 2 / 3 / 1
2 / 3 / 4 / 3
3 / 5 / 5 / 6
(b) Discuss the concepts: (i) sub optimality and (ii) curse of dimensionality for DP. (06)
Q.5.(a) What do you understand by local and global extremities? Discuss convexity of a function. (09)
(b) Find the extremities of the function (06) Z = 2X1 + X2 + 10 Subject to the constraint X1 +2X22 = 3
Q.6 (a)Explain Transportation array and discuss its noteworthy features. Describe steps in North West Corner method for obtaining initial feasible solution. (08)
(b) Total available supply at each of the three Origins namely A, B and C is 7, 9 and 18 respectively. The total demand at each of the four destinations is given as 5, 8, 7 and 14 respectively. Obtain initial feasible solution using North West Corner Rule. (07)
/ SGGS Institute of Engg. and Tech., NandedAcademic Year 2013-14 (Semester II)
Class: M. Tech. (Civil & WM Engg.) Summer Term
Subject: Systems Engineering & Its Applications (MCW512)
Date: 07/07/2014 / Time: 14.30 – 17.30. / Max. Marks : 100
Notes:(i) Solve any five questions from Q.1 to Q.6.
(ii) Solve any one question from Q.7Q.8.
(iii) Q. 9 is compulsory.
Q.1 (a) What are different forms of LP Model? State the related equations & the associated terms. (08)
(b) What do you mean by basic variables and non-basic variables? (04)
(c) ) What are various types of constraints? (04)
Q.2 (a) How do you obtain the standard LP formulation from the general LP using Slack variable,
Surplus variable and artificial variables? (06)
(b) Describe the steps in graphical method of solution for a LP model (06)
(c) What are different types of solutions that can be obtained for a LP model? (04)
Q.3 (a) State the general form of a NLP Model and explain the general solution methodology
for unconstrained NLP problem (Hessian Matrix method). (10)
(b) Describe the Kuhn-Tucker Conditions for inequality constrained NLP formulation. (06)
Q.4(a) State the requirements of DP formulation.Explain the Bellman's principle of optimality. (09)
(b) State the DP model for a resource allocation problem and explain the terminology. (07)
Q.5(a) Describe briefly the transportation problem and state the equations.State general structure
characteristics of Transportation Array. (10)
(b) Explain North West Corner and least cost methods for obtaining initial feasible solution of
the transportation problem (06)
Q.6 Write short notes on (any four) : (16)
(i) Curse of dimensionality (ii) Canonical form and its use
(iii) Duality in LP (iv) Method of direct substitution (v) convexity of a function
Q.7 Solve : Maximize f = 3X1 + 2X2 + X3 + 4X4 (10)
Subject to : 2X1 + 2X2 + X3 + 3X4 ≤ 20
3X1 + X2 + 2X3 + 2X4 ≤ 20 and Non negativity constraint
Q. 8 Find the extremities of the function and give your comments (10) Z = 2X1 + X2 + 10 Subject to the constraint X1 +2X22 = 3
Q. 9 (a)Solve the following LP formulations graphically. (12)
Maximize z = 6x + 4ysubject to
1.5 x + 3 y 3
4 x + 2 y 12
and x, y 0 / Maximize z = 2x + 1.5 y
subject to
2x + 3y 6
2x + 3y 12
and x, y 0
(b) Total available supply at each of the three Origins is 21, 27 and 54 respectively. The total demand at each of the four destinations is given as 15, 24, 21 and 42 respectively. Obtain initial feasible solution using North West Corner Rule. (08)
/ SGGS Institute of Engg. and Tech., NandedAcademic Year 2012-13
Class: M. Tech. (Civil & WM Engg.) Summer Term Examination
Subject: Systems Engineering& its applications (MCW 512)
Date: 3rd July, 2013 / Time: 10.00 to 13.00 hrs / Max. Marks : 100
Q. 1. Solve any FIVE of the following:(40)
(i) Define system and state its characteristics. Enlist various techniques of systems analysis.
(ii) State the requirements of DP and Explain the Bellman’s principle of optimality.
(iii) What are various forms of LP model? State relevant equations and explain the terms.
(iv) Discuss briefly the transformations required to obtain the standard form of LP.
(v) State the standard NLP formulation. Compare briefly Langrange Multiplier method with the Direct Substitution method of solution.
(vi) Discuss the Kuhn-Tucker conditions for Non-linear programming problem.
Q. 2.Solve the following LP formulations graphically.(14)
Maximize z = 3x + 2ysubject to
x + 2y 2
2x + y 6
and x, y 0 / Maximize z = 4x + 3y
subject to
2x + 3y 6
4x+ 6y24
and x, y 0
Q.3. (a) Describe the transportation problem with mathematical formulation. How do you write the corresponding transportation array? (07)
(b)solve the following transportation array by North West corner rule and least cost method. (no. in cells indicate unit cost and Total indicates demand/supply) (09)
D1 / D2 / D3 / D4 / D5 / TotalO1 / 12 / 4 / 9 / 5 / 9 / 55
O2 / 8 / 1 / 6 / 6 / 7 / 45
O3 / 1 / 12 / 4 / 7 / 7 / 30
O4 / 10 / 15 / 6 / 9 / 1 / 50
Total / 40 / 20 / 50 / 30 / 40
Q. 4. Solve the following LP problem.(10)
Maximize Z = 2X1 + X2
Subject to3X1 + X2≤ 300
4X1 + 2X2≤ 500.
Q. 5. (a) State the composition of Hessian Matrix. How do you decide its nature?(05)
(b) Discuss the concavity and convexity of a function.(05)
Q.6.Explain Local and Global Extremities. Find the extremities of the function(10)
Z = 5X12 + 2X2 – X1X2
Subject to X1 + X2 = 3
(iii) State the various forms of LP formulations and explain the terms. Explain briefly the simplex method and state the limitations.
Explain the Hessian Matrix. How do you decide its nature? Illustrate.
Discuss briefly the transformations required to obtain the standard form of LP.
(i)
Q. 3. (a) State primal-dual relationships. Write the Dual the following LP
Minimize Z = 2X1 + X2(11)
Subject to : 3X1 + X2≥ 3
X1 + 2X2≥ 3
4X1 + 3X2≥ 6
(a) What do you understand by local and global extremities? Discuss convexity of a function. (09)