Source document. This page reproduces the syllabus this course was written to, as published — its semesters, credits and paper numbers are that document’s, not this site’s. The course itself is studied on its own, in any order.

Course Information

Title	Optimization Techniques
Theory Credits	3 (3 hrs/week)
Practical Credits	1 (2 hrs/week)

Course Outcomes

1. Solve problems in logistics through transportation models.
2. Find solutions to problems having space / capacity constraints.
3. Minimise total elapsed time in industry by efficient allocation of jobs to suitable persons.
4. Find solutions for adequate usage of human resources via assignment.
5. Find the most plausible solutions in industries and agriculture when a random environment exists, using game theory and network analysis.

Theory — Five Units

Unit 1: Transportation Problem

Introduction, mathematical formulation. Initial Basic Feasible Solution — North-West Corner rule, Lowest-Cost Entry method, Vogel's Approximation Method. Method of finding optimal solution — MODI method. Degeneracy in transportation problem, resolution of degeneracy, unbalanced transportation problem. Maximization of TP.

Open Unit 1 →

Unit 2: Assignment Problem

Introduction, mathematical formulation. Hungarian method for solving assignment problems — both balanced and unbalanced cases. Maximization. Restricted assignment problems.

Open Unit 2 →

Unit 3: Sequencing Problem

Introduction and assumptions of sequencing problems. Johnson's algorithm and problems for n jobs on two machines. Algorithm and problems for n jobs on three machines. Algorithm and problems for n jobs on m machines.

Open Unit 3 →

Unit 4: Game Theory

Two-person zero-sum games. Pure and mixed strategies. Maxmin and Minimax principles, saddle point and its existence. Games without saddle point. Mixed strategies. Solution of 2×2 rectangular games. Graphical method of solving 2×n and m×2 games. Dominance property.

Open Unit 4 →

Unit 5: Network Scheduling

Basic components of a network — nodes and arcs, events and activities. Rules of network construction. Time calculations in networks. Critical Path Method (CPM) and Programme Evaluation and Review Technique (PERT).

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Practical — List of Experiments (8)

1. IBFS of transportation problem using NWCM, LCEM and VAM.
2. Optimum solution to balanced and unbalanced transportation problems by MODI method.
3. Solution of assignment problem using Hungarian method (minimization cases).
4. Solution of sequencing problem — n jobs through two machines.
5. Solution of sequencing problem — n jobs through three machines.
6. Project scheduling of a given project (deterministic case — CPM).
7. Project scheduling of a given project (probabilistic case — PERT).
8. Solution of m × n games by dominance rule.

Open practical course material →

Text Books / References

1. S. D. Sharma — Operations Research, Kedar Nath Ram Nath & Co., Meerut.
2. Kanti Swarup, P. K. Gupta & Man Mohan — Operations Research, Sultan Chand & Sons.
3. J. K. Sharma — Operations Research and Application, Macmillan, New Delhi.
4. Gass S. I. — Linear Programming, McGraw-Hill.
5. Hadley G. — Linear Programming, Addison-Wesley.
6. H. M. Taha — Operations Research: An Introduction, Macmillan.

Suggested Co-Curricular Activities

1. Training by related industrial experts (logistics, operations managers).
2. Assignments and case studies of real industrial problems.
3. Seminars, group discussions and quizzes on OR techniques.
4. Computational labs with Excel Solver, R or Python (PuLP).
5. Field visits to manufacturing facilities, ports or logistics hubs.
6. Invited lectures by OR practitioners.

Exam Blueprint (recommended)

Unit	Topic	Approx. Weightage
1	Transportation Problem	25 %
2	Assignment Problem	15 %
3	Sequencing Problem	15 %
4	Game Theory	20 %
5	Network Scheduling (CPM/PERT)	25 %

Quick Reference — Key Methods & Formulas

Topic	Method / Formula
TP — IBFS	NWCM (easiest), LCEM (cost-aware), VAM (most accurate)
TP — Optimality	MODI: \(u_i + v_j = c_{ij}\) on basic cells; \(d_{ij} = c_{ij} - (u_i + v_j) \ge 0\) ⇒ optimum
TP — Maximization	Replace \(c_{ij}\) by \(M - c_{ij}\); minimize
Assignment — Hungarian	Row red → Col red → Cover lines (k) → if k < n adjust by min uncovered
Sequencing — n × 2	Johnson's rule: min on M₁ → schedule early; min on M₂ → schedule late
Sequencing — n × 3	If min(M₁) ≥ max(M₂) or min(M₃) ≥ max(M₂): use G = M₁+M₂, H = M₂+M₃
Game — Saddle	Saddle exists iff Maxmin = Minimax; common entry = value
2×2 mixed	\(p = (d-c)/[(a+d)-(b+c)]\); \(v = (ad-bc)/[(a+d)-(b+c)]\)
Dominance	Row Aᵢ uniformly ≥ Aₖ ⇒ Aₖ dominated, delete
CPM — Forward	\(E_j = \max_{i \to j}(E_i + t_{ij})\)
CPM — Backward	\(L_i = \min_{i \to j}(L_j - t_{ij})\)
CPM — Total Float	\(\text{TF}_{ij} = L_j - E_i - t_{ij}\)
Critical Path	Activities with TF = 0
PERT — Expected time	\(t_e = (a + 4m + b)/6\)
PERT — Variance	\(\sigma^2 = ((b - a)/6)^2\)
PERT — Probability	\(P(T_p \le T) = \Phi((T - T_e)/\sigma_{\text{project}})\)