🧱 Stepping Stone Method – Transportation Problems (CA Corporate Level)

📘 What is the Stepping Stone Method?

The Stepping Stone Method is a technique used to optimize a basic feasible solution of a transportation problem. It helps determine whether the current solution is optimal and identifies unused routes that can potentially reduce total transportation cost.

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🎯 Objectives

• 🚚 Optimize transportation cost

• 📦 Evaluate unused (empty) routes

• 🔁 Adjust current allocations for improvement

• ✅ Move toward the optimal solution

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🧮 Prerequisites

• A balanced transportation table (total supply = total demand)

• An initial basic feasible solution (IBFS) using methods like:

  • 🟨 Northwest Corner Method

  • 📉 Least Cost Method

  • 🔄 Vogel’s Approximation Method (VAM)

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📐 Stepping Stone Method – Step-by-Step

Step 1: 🔍 Identify Unused Routes

Pick one unused (empty) cell in the transportation table.

Step 2: 🔁 Trace a Closed Path

Form a loop (closed path) starting and ending at the selected cell, moving horizontally and vertically (not diagonally) through allocated cells only.

Step 3: ➕➖ Assign Signs

Alternate + and – signs on each corner of the loop, starting with + at the unused cell.

Step 4: 💰 Calculate Net Cost Change

For each unused cell:

$$\Delta C = \sum (\text{Cost at '+' cells}) - \sum (\text{Cost at '–' cells})$$

If ΔC < 0 → it means cost can be reduced by reallocating.

Step 5: 🔧 Reallocate Shipments

If ΔC is negative, adjust the allocations along the loop:

• Add units to '+' cells

• Subtract same units from '–' cells

• The smallest allocation in '–' cells becomes the limiting factor

Step 6: 🔁 Repeat

Repeat the process until all ΔC ≥ 0 for all unused cells → then the solution is optimal.

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📊 Example: Stepping Stone Method

Let’s consider a balanced transportation table with:

🚛 Supply & Demand:

	D1	D2	D3	Supply
S1	4	6	8	20
S2	2	3	6	30
S3	1	5	7	25
Demand	30	25	20

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🔢 Step A: Initial Basic Feasible Solution using Least Cost Method

	D1	D2	D3	Supply
S1	4 (0)	6 (0)	8 (20)	0
S2	2 (30)	3 (0)	6 (0)	0
S3	1 (0)	5 (25)	7 (0)	0
Demand	0	0	0

Total Cost = (20×8) + (30×2) + (25×5) = ₹160 + ₹60 + ₹125 = ₹345

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🔍 Step B: Evaluate Unused Cell (S3-D1)

Loop:

• Start at S3-D1 (+)

• Move right to S2-D1 (–)

• Down to S2-D3 (+)

• Left to S3-D3 (–)

Loop Costs:

• S3-D1 = 1 (+)

• S2-D1 = 2 (–)

• S2-D3 = 6 (+)

• S3-D3 = 7 (–)

$$\Delta C = (1 + 6) - (2 + 7) = 7 - 9 = -2$$

Since ΔC is negative → there’s room for improvement

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🔧 Step C: Reallocate Along Loop

Adjust minimum units from '–' cells. The minimum value is 0, so no reallocation possible for this path.

Repeat the same for other unused cells.

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✅ Optimality Check

When all ΔC values for unused routes ≥ 0, the solution is optimal.

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📝 Conclusion

The Stepping Stone Method is a systematic tool to improve the transportation cost in logistics and cost accounting. For CA Corporate Level, it’s a critical technique under Operations Research and Decision Analysis.

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📌 Quick Recap

Concept	Description
Loop Formation	Closed path through allocated cells
EMV Formula	ΔC = Total (+) cost – Total (–) cost
Improvement Decision	If ΔC < 0 → Improve allocation
Stop When	All ΔC ≥ 0 (Optimal solution)

📚 Explore More Notes:

• Transportation Problem (Basics)

• Vogel’s Approximation Method (VAM)

• Northwest Corner Rule Explained

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Tags: #SteppingStoneMethod #TransportationProblem #Optimization #CACorporateLevel #CostAccounting #OperationsResearch #Comztube

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