Syllabus (MATHEMATICS)

Course Type: MAJ-18

Semester: 8

Course Code: BMTMMAJ18T

Course Title: Operations Research & Calculus of Variations

(L-P-Tu): 5-0-1

Credit: 6

Practical/Theory: Theory

Course Objective: To be done.

Learning Outcome: Course Outcomes (CO): The whole course will have the following outcomes: Upon successful completion, the students will be able to CO1: formulate Network models for service and manufacturing systems, and apply operations research techniques and algorithms

Syllabus:

Unit -1: Operations Research [Credit: 3, 45L]

• Replacement Problems: Introduction, Replacement policies for items whose efficiency deteriorates with time, Individual and group replacement, Replacement policies for items that fail completely.
• Job sequences: Processing of n jobs through two machines, The Algorithm, Processing of n jobs through m machines.
• Project Network: Introduction, Basic differences between PERT and CPM, Steps of PERT/CPM Techniques, PERT/CPM network Components and Precedence Relationships, Critical Path analysis, Probability in PERT analysis, Project Crashing, Time cost Trade-off procedure, Updating of the Project.
• Flow Network: Max-flow min-cut theorem, Generalized Max flow min-cut theorem, Linear Programming interpretation of Max-flow min-cut theorem, Minimum cost flows, Min-flow max-cut theorem.
• Inventory control Models: Classification of Inventories, Advantage of Carrying Inventory, Features of Inventory System, Deterministic inventory models including price breaks, Discrete and Continuous probabilistic inventory models, Safety stock and Buffer Stock, Concept of just in time inventory
• Queuing Theory: Introduction, characteristic of Queuing systems, operating characteristics of Queuing system. Probability distribution in Queuing systems. Classification of Queuing models. Poisson and non-Poisson queuing models (M/M/1:∞/FCFS/∞), (M/M/C:∞/FCFS/∞), (M/M/1:N/FCFS/∞), (M/M/C:N/FCFS/∞).

Unit -2: Calculus of Variations [Credit: 2, 30L]

Variation, Linear functional, Deduction of Euler-Lagrange differential equation and some special cases of it, Euler-Lagrange differential equation for multiple dependent variables, Functional dependent on higher order derivatives, Functional dependent on functions of several variables. Application of Calculus of variations for the problems of shortest distance, minimum surface of revolution, Brachistochrone problem, geodesic, Isoperimetric problem, Calculus of variations for problems in parametric form, Variational problems with moving boundaries. Sturm-Liouville problems, Hamilton’s principle, Lagrange’s equations, Generalized dynamical entities, vibrating string, vibrating membranes, theory of elasticity – The variational problem of a vibrating elastic plate.

Reading References:

1. J. K. Sharma; Operations Research (Theory and Applications); Macmillan.
2. A. K. Bhunia, Shri L. Sahoo; Advanced Operations Research; Asian Books Pvt. Ltd.
3. J. D. Weist, F. K. Levy: A Management Guide to PERT/ CPM. 2nd Edition, PHI, 1967 (Reprint 2007).
4. Ronald L. Rardin: Optimization in Operations Research, Prentice Hall, 1998.
5. Hamdy A. Taha: Operations Research-An Introduction, Prentice Hall, 9th Edition, 2010.
6. Donald Gross, John F. Shortle, James M. Thompson, Carl M. Harris: Fundamentals of Queueing Theory, 4th Edition, 2008.
7. Donald Waters: Inventory Control and Management, John Wiley, 2010.
8. R Weinstock, Calculus of Variation, Dover, 1970.
9. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover Publications, 2000
10. Advanced Engineering Mathematics, E Kreyszig, John Wiley and Sons, Ninth Edition, 2012
11. A. S. Gupta, Calculus of Variations with Applications, Prentice Hall of India, 1997
12. Gelfand, J.M., Fomin, S.V., Calculus of Variations, Prentice Hall, New Jersey, 1963.