It's about transportation formulation and solutions

1. Transportation Problem in Operations
Research
Optimizing Logistics and Costs
Presented by: [Kavya Sagar]

2. Introduction
• A special type of Linear Programming Problem (LPP).
• Involves transporting goods from sources to destinations at
minimal cost.
• Essential in logistics, supply chains, and resource allocation.

3. Problem Definition
• Objective: Minimize transportation cost.
• Constraints: Meet supply and demand at each node.
• Components: Sources, Destinations, and Cost Matrix.

4. Mathematical Formulation
• Minimize: Z = ΣΣ cᵢ * xᵢ
ⱼ ⱼ
• Subject to:
• Σ xᵢ (over j) = sᵢ (Supply constraint)
ⱼ
• Σ xᵢ (over i) = d (Demand constraint)
ⱼ ⱼ
• xᵢ ≥ 0
ⱼ

5. Initial Solution Methods
• 1. North-West Corner Rule (NWCR)
• 2. Least Cost Method (LCM)
• 3. Vogel’s Approximation Method (VAM)

6. Example Problem: Cost Table (Balanced)
D1 (20) D2 (30) D3 (25) D4 (50)
S1 8 6 10 0
S2 9 12 13 0
S3 14 9 16 0

7. North-West Corner Rule (NWCR)
• Allocations:
• S1→D1 = 20, S1→D2 = 30
• S2→D3 = 25, S2→D4 = 15
• S3→D4 = 35
•Total Cost = 665
D1 (20) D2 (30) D3 (25) D4 (50) Supply
S1 8 6 10 0 50
S2 9 12 13 0 40
S3 14 9 16 0 35
Demand 20 30 25 50 125

8. Least Cost Method (LCM)
• Allocations:
• • S1→D4 = 50, S2→D2 = 30, S2→D1 = 10
• • S3→D1 = 10, S3→D3 = 25
• Total Cost = 990

9. Vogel’s Approximation Method (VAM)
• Allocations:
• • S2→D4 = 40, S3→D4 = 10
• • S1→D2 = 30, S1→D1 = 20
• • S3→D3 = 25
• Total Cost = 740

10. Real-Life Applications
• Logistics and Freight Distribution
• Humanitarian Aid Logistics
• Retail Chain Supply Distribution
• Healthcare Delivery Networks

11. Advantages and Limitations
Advantages:
• Minimizes cost
• Streamlines decision-making
Limitations:
• Assumes constant unit costs
• Doesn’t consider route delays or dynamic changes

12. Conclusion
• Transportation problem aids cost optimization in logistics.
• Initial solutions using NWCR, LCM, and VAM are foundational.
• Useful in real-world decision-making across industries.

13. References
• Operations Research by Kanti Swarup
• Taha: Operations Research – An Introduction
• NPTEL, ResearchGate, MIT OpenCourseWare