Answer:

Definition:

Economic Order Quantity (EOQ) is the optimal order quantity that minimizes the total inventory costs, which include ordering costs and carrying costs. It represents the most economical quantity to order at one time to achieve the lowest total cost of inventory management.

Importance in Inventory Management:

1. Cost Optimization: EOQ helps minimize the total cost by balancing ordering and holding costs. For example, if a company orders too frequently (small quantities), ordering costs increase. If it orders infrequently (large quantities), carrying costs increase. EOQ finds the perfect balance.
2. Improved Cash Flow: By ordering the right quantity, companies avoid tying up excessive capital in inventory. Example: A retailer using EOQ of 500 units instead of bulk ordering 5,000 units frees up cash for other business operations.
3. Better Planning: EOQ provides a systematic approach to inventory decisions, making procurement planning more predictable and efficient. Example: A manufacturer can schedule orders based on EOQ, ensuring smooth production without interruptions.
4. Storage Efficiency: Prevents overstocking and reduces warehouse space requirements. Example: A pharmaceutical company saves on cold storage costs by maintaining optimal inventory levels.

Derivation of EOQ Formula:

Let:

• Q = Order quantity (units)
• A = Annual demand (units/year)
• O = Ordering cost per order (₹)
• C = Carrying cost per unit per year (₹)

Total Annual Cost = Ordering Cost + Carrying Cost

Ordering Cost = (Number of orders per year) × (Cost per order)

Number of orders = A/Q

Therefore, Ordering Cost = (A/Q) × O

Carrying Cost = (Average inventory) × (Carrying cost per unit)

Average inventory = Q/2

Therefore, Carrying Cost = (Q/2) × C

Total Cost (TC) = (A/Q) × O + (Q/2) × C

To minimize total cost, we differentiate with respect to Q and set equal to zero:

dTC/dQ = -AO/Q² + C/2 = 0

C/2 = AO/Q²

Q² = 2AO/C

Q = √(2AO/C) = EOQ

Example: If A = 10,000 units, O = ₹500, C = ₹20, then EOQ = √(2×10,000×500/20) = 707 units

Answer:

Given Data:

• Annual Demand (A) = 18,000 units
• Ordering Cost (O) = ₹900 per order
• Carrying Cost (C) = ₹50 per unit per year
• Lead Time = 10 days

(i) Economic Order Quantity (EOQ):

EOQ = √(2 × A × O / C)

EOQ = √(2 × 18,000 × 900 / 50)

EOQ = √(32,400,000 / 50)

EOQ = √648,000

EOQ = 805.0 units (approximately 805 units)

(ii) Number of Orders Per Year:

Number of orders = Annual Demand / EOQ

Number of orders = 18,000 / 805

Number of orders = 22.36 ≈ 22 orders per year

(iii) Time Between Orders:

Time between orders = 365 days / Number of orders

Time between orders = 365 / 22.36

Time between orders = 16.32 days (approximately every 16 days)

(iv) Total Annual Inventory Cost:

Total Cost = Ordering Cost + Carrying Cost

Ordering Cost = Number of orders × Cost per order

Ordering Cost = 22.36 × 900 = ₹20,124

Carrying Cost = (EOQ/2) × Carrying cost per unit

Carrying Cost = (805/2) × 50 = 402.5 × 50 = ₹20,125

Total Cost = ₹20,124 + ₹20,125

Total Annual Inventory Cost = ₹40,249

(v) Reorder Point:

Daily Demand = Annual Demand / 365 days

Daily Demand = 18,000 / 365 = 49.32 units per day

Reorder Point = Daily Demand × Lead Time

Reorder Point = 49.32 × 10

Reorder Point = 493 units

Interpretation: The company should order 805 units each time, place orders approximately 22 times per year (every 16 days), with a total inventory cost of ₹40,249. When inventory level reaches 493 units, a new order should be placed to account for the 10-day lead time.

Answer:

ASSUMPTIONS OF EOQ MODEL:

1. Constant Demand Rate: The model assumes that demand is known and constant throughout the year. Example: 10,000 units per year means approximately 27-28 units per day consistently.
2. Instantaneous Replenishment: It assumes that when an order is placed, the entire quantity arrives at once with no delivery delays.
3. No Stockouts: The model assumes that demand can always be met, and there are no stockout situations.
4. Constant Costs: Ordering costs and carrying costs remain constant regardless of order size or quantity.
5. No Quantity Discounts: The purchase price per unit remains same regardless of order quantity.
6. Single Product: EOQ is calculated for one product at a time, assuming independent inventory items.
7. Infinite Planning Horizon: The model assumes continuous operation without any termination point.

LIMITATIONS OF EOQ MODEL:

1. Demand Uncertainty: In reality, demand fluctuates due to seasonality, market conditions, and consumer preferences. Example: Ice cream sales vary by season, making constant demand assumption unrealistic.
2. Lead Time Variability: Suppliers may not always deliver on time due to various factors like transportation issues, production delays, or natural disasters.
3. Bulk Discounts Ignored: Many suppliers offer quantity discounts which are not considered in basic EOQ. Example: 5% discount on orders above 1,000 units might make ordering more than EOQ economical.
4. Storage Constraints: Physical warehouse capacity limitations may prevent ordering the calculated EOQ quantity.
5. Capital Constraints: Companies may not have sufficient working capital to order the EOQ quantity.
6. Multiple Products: Real businesses manage hundreds of products, and EOQ doesn't account for interdependencies or shared resources.

ADDRESSING LIMITATIONS IN PRACTICE:

Limitation	Practical Solution
Variable Demand	Use EOQ with safety stock; Apply probabilistic models; Adjust EOQ quarterly based on demand forecasts
Quantity Discounts	Modify EOQ to compare total costs at different price break points; Use EOQ with price breaks formula
Storage Limitations	Consider warehouse capacity as a constraint; Use EOQ as a guideline and adjust to practical maximum
Lead Time Uncertainty	Maintain safety stock; Use reorder point = (Average demand × Lead time) + Safety stock
Multiple Products	Apply ABC analysis; Calculate EOQ for Class A items; Use periodic review for Class C items

Conclusion: While EOQ has limitations, it remains a valuable tool when used with appropriate modifications and as part of a comprehensive inventory management system that includes safety stock, reorder points, and regular review mechanisms.

Answer:

Given Data:

• Annual Demand (A) = 15,000 units
• Purchase Price (P) = ₹200 per unit
• Ordering Cost (O) = ₹750 per order
• Carrying Cost = 20% of purchase price
• Discount = 5% on orders ≥ 1,500 units

Step 1: Calculate Carrying Cost per unit

Carrying Cost (C) = 20% of ₹200 = 0.20 × 200 = ₹40 per unit per year

Step 2: Calculate EOQ (without considering discount)

EOQ = √(2 × A × O / C)

EOQ = √(2 × 15,000 × 750 / 40)

EOQ = √(22,500,000 / 40)

EOQ = √562,500

EOQ = 750 units

Step 3: Calculate Total Cost at EOQ (750 units)

Purchase Cost:

Purchase Cost = Annual Demand × Price per unit

Purchase Cost = 15,000 × 200 = ₹30,00,000

Ordering Cost:

Number of orders = 15,000 / 750 = 20 orders

Ordering Cost = 20 × 750 = ₹15,000

Carrying Cost:

Carrying Cost = (750/2) × 40 = ₹15,000

Total Cost at EOQ:

Total = ₹30,00,000 + ₹15,000 + ₹15,000

Total Cost at EOQ = ₹30,30,000

Step 4: Calculate Total Cost with Discount (Order quantity = 1,500 units)

Discounted Price:

Discount = 5% of ₹200 = ₹10

New Price = ₹200 - ₹10 = ₹190 per unit

New Carrying Cost:

New C = 20% of ₹190 = ₹38 per unit per year

Purchase Cost:

Purchase Cost = 15,000 × 190 = ₹28,50,000

Ordering Cost:

Number of orders = 15,000 / 1,500 = 10 orders

Ordering Cost = 10 × 750 = ₹7,500

Carrying Cost:

Carrying Cost = (1,500/2) × 38 = ₹28,500

Total Cost with Discount:

Total = ₹28,50,000 + ₹7,500 + ₹28,500

Total Cost with Discount = ₹28,86,000

Step 5: Comparison and Decision

Scenario	Order Quantity	Total Annual Cost	Savings
Without Discount (EOQ)	750 units	₹30,30,000	-
With 5% Discount	1,500 units	₹28,86,000	₹1,44,000

Conclusion:

YES, the retailer should take advantage of the 5% discount by ordering 1,500 units per order. This will result in annual savings of ₹1,44,000 (4.75% reduction in total cost) despite ordering double the EOQ quantity. The savings from reduced purchase price outweigh the increased carrying costs.

Answer:

RELATIONSHIP BETWEEN EOQ, ORDERING COSTS, AND CARRYING COSTS:

The fundamental relationship is expressed in the EOQ formula:

EOQ = √(2 × A × O / C)

Key observations:

1. Direct Relationship with Ordering Cost (O): EOQ is directly proportional to the square root of ordering cost. Higher ordering costs lead to larger order quantities to reduce the frequency of orders.
2. Inverse Relationship with Carrying Cost (C): EOQ is inversely proportional to the square root of carrying cost. Higher carrying costs encourage smaller order quantities to reduce inventory holding expenses.
3. Optimal Balance Point: At EOQ, total ordering cost equals total carrying cost, achieving the minimum total inventory cost.

MATHEMATICAL ANALYSIS OF CHANGES:

Base Case Example:

• A = 10,000 units
• O = ₹500
• C = ₹20
• EOQ = √(2 × 10,000 × 500 / 20) = √500,000 = 707 units

(a) When Ordering Cost Doubles (O becomes 2O):

New EOQ = √(2 × A × 2O / C)

New EOQ = √(2 × 2 × A × O / C)

New EOQ = √2 × √(2 × A × O / C)

New EOQ = √2 × Original EOQ

New EOQ = 1.414 × Original EOQ

Numerical Example:

New O = 2 × 500 = ₹1,000

New EOQ = √(2 × 10,000 × 1,000 / 20) = √1,000,000 = 1,000 units

Increase = 1,000 / 707 = 1.414 (41.4% increase)

Conclusion: When ordering cost doubles, EOQ increases by approximately 41.4% (factor of √2)

(b) When Carrying Cost Increases by 50% (C becomes 1.5C):

New EOQ = √(2 × A × O / 1.5C)

New EOQ = √(1/1.5) × √(2 × A × O / C)

New EOQ = 1/√1.5 × Original EOQ

New EOQ = 0.816 × Original EOQ

Numerical Example:

New C = 20 + (50% of 20) = ₹30

New EOQ = √(2 × 10,000 × 500 / 30) = √333,333 = 577 units

Change = 577 / 707 = 0.816 (18.4% decrease)

Conclusion: When carrying cost increases by 50%, EOQ decreases by approximately 18.4%

(c) When Annual Demand Triples (A becomes 3A):

New EOQ = √(2 × 3A × O / C)

New EOQ = √3 × √(2 × A × O / C)

New EOQ = √3 × Original EOQ

New EOQ = 1.732 × Original EOQ

Numerical Example:

New A = 3 × 10,000 = 30,000 units

New EOQ = √(2 × 30,000 × 500 / 20) = √1,500,000 = 1,225 units

Increase = 1,225 / 707 = 1.732 (73.2% increase)

Conclusion: When annual demand triples, EOQ increases by approximately 73.2% (factor of √3)

SUMMARY TABLE:

Change in Variable	Effect on EOQ	Mathematical Factor	Example Change
Ordering Cost doubles	EOQ increases	× √2 = × 1.414	707 → 1,000 units (+41.4%)
Carrying Cost increases 50%	EOQ decreases	× 0.816	707 → 577 units (-18.4%)
Annual Demand triples	EOQ increases	× √3 = × 1.732	707 → 1,225 units (+73.2%)

KEY INSIGHTS:

• EOQ is more sensitive to changes in demand than to changes in costs due to square root relationship
• Increases in ordering costs lead to larger order quantities (order less frequently)
• Increases in carrying costs lead to smaller order quantities (order more frequently)
• The square root relationship provides a cushioning effect—large changes in parameters result in proportionally smaller changes in EOQ

Answer:

Given:

• Annual Demand (A) = 8,000 units
• Ordering Cost (O) = ₹400 per order
• Carrying Cost (C) = ₹25 per unit per year

Formula:

EOQ = √(2 × A × O / C)

Calculation:

EOQ = √(2 × 8,000 × 400 / 25)

EOQ = √(6,400,000 / 25)

EOQ = √256,000

EOQ = 506 units (approximately)

Interpretation: The company should order 506 units each time to minimize total inventory costs. This will result in approximately 16 orders per year (8,000 ÷ 506).

Answer:

Six Advantages of EOQ:

1. Cost Minimization: EOQ minimizes total inventory costs by optimally balancing ordering costs and carrying costs, leading to significant savings in inventory management expenses.
2. Improved Cash Flow: By preventing overstocking, EOQ frees up working capital that would otherwise be tied up in excess inventory, allowing better utilization of financial resources.
3. Better Planning: EOQ provides a systematic and predictable approach to inventory management, making procurement planning more efficient and reliable.
4. Reduced Storage Requirements: By ordering optimal quantities, companies avoid excessive inventory accumulation, reducing warehousing costs and space requirements.
5. Prevention of Stockouts: Systematic reordering based on EOQ helps maintain adequate stock levels, preventing production disruptions and lost sales due to stock unavailability.
6. Scientific Decision Making: EOQ provides a mathematical and logical basis for inventory decisions, replacing guesswork with quantitative analysis and enabling management to make data-driven decisions.

Answer:

Given:

• EOQ = 600 units
• Annual Demand (A) = 7,200 units
• Carrying Cost (C) = ₹30 per unit per year
• Ordering Cost (O) = ?

Formula Rearrangement:

Starting with: EOQ = √(2 × A × O / C)

Squaring both sides: EOQ² = 2 × A × O / C

Rearranging for O: O = (EOQ² × C) / (2 × A)

Calculation:

O = (600² × 30) / (2 × 7,200)

O = (360,000 × 30) / 14,400

O = 10,800,000 / 14,400

O = ₹750 per order

Verification:

EOQ = √(2 × 7,200 × 750 / 30) = √(10,800,000 / 30) = √360,000 = 600 units ✓

Answer:

Components of Total Inventory Cost:

1. Ordering Cost (O):

These are costs incurred each time an order is placed, regardless of order size. Ordering costs are fixed per order.

Components include:

• Purchase requisition and order preparation costs
• Communication with suppliers (phone, email, visits)
• Order processing and verification
• Transportation and shipping charges
• Receiving, inspection, and quality checking

Example: If it costs ₹500 to process each order (₹200 for paperwork, ₹150 for communication, ₹150 for receiving inspection), then O = ₹500 per order.

2. Carrying Cost (C):

These are costs associated with holding inventory over time. Carrying costs are variable per unit.

Components include:

• Warehouse rental and utilities
• Insurance premiums on inventory
• Obsolescence and deterioration
• Opportunity cost of capital (interest on funds tied up)
• Handling and material management costs
• Theft, damage, and shrinkage

Example: If annual warehouse cost is ₹10/unit, insurance is ₹5/unit, and capital cost is ₹15/unit, then C = ₹30 per unit per year.

Total Inventory Cost Formula:

Total Cost = (A/Q) × O + (Q/2) × C

Where Q is order quantity and A is annual demand.

Example Calculation: If A = 10,000 units, EOQ = 700 units, O = ₹500, C = ₹20:

• Ordering Cost = (10,000/700) × 500 = ₹7,143
• Carrying Cost = (700/2) × 20 = ₹7,000
• Total Cost = ₹14,143

Answer:

Given:

• Annual Demand (A) = 6,000 units
• EOQ = 500 units
• Ordering Cost (O) = ₹600 per order

Part 1: Calculate Carrying Cost per unit per year

Using: EOQ = √(2 × A × O / C)

Squaring: EOQ² = 2 × A × O / C

Solving for C: C = (2 × A × O) / EOQ²

C = (2 × 6,000 × 600) / (500)²

C = 7,200,000 / 250,000

C = ₹28.80 per unit per year

Part 2: Calculate Total Annual Inventory Cost

Ordering Cost:

Number of orders = A / EOQ = 6,000 / 500 = 12 orders

Ordering Cost = 12 × 600 = ₹7,200

Carrying Cost:

Average Inventory = EOQ / 2 = 500 / 2 = 250 units

Carrying Cost = 250 × 28.80 = ₹7,200

Total Annual Inventory Cost:

Total Cost = ₹7,200 + ₹7,200

Total Cost = ₹14,400

Note: At EOQ, ordering cost equals carrying cost, which validates our calculation (both are ₹7,200).

Answer:

Economic Order Quantity (EOQ) is the optimal order quantity that a company should purchase to minimize its total inventory costs, including both ordering costs and carrying costs. It represents the ideal order size that balances the cost of placing orders with the cost of holding inventory.

EOQ = √(2 × A × O / C)

Where A = Annual Demand, O = Ordering Cost, C = Carrying Cost per unit per year.

Answer:

EOQ Formula:

EOQ = √(2 × A × O / C)

Components:

• A (Annual Demand): Total quantity required per year (in units)
• O (Ordering Cost): Cost incurred per order placement (in rupees)
• C (Carrying Cost): Cost of holding one unit in inventory for one year (in rupees per unit per year)

Answer:

Ordering Costs are costs incurred each time an order is placed for inventory, regardless of the order size. These are fixed costs per order.

Examples of Ordering Costs:

• Purchase requisition preparation costs
• Vendor communication expenses (phone, email)
• Order processing and approval costs
• Transportation and freight charges
• Receiving, inspection, and quality control costs

Answer:

Given: A = 5,000, O = ₹250, C = ₹10

Formula: EOQ = √(2 × A × O / C)

Calculation:

EOQ = √(2 × 5,000 × 250 / 10)

EOQ = √(2,500,000 / 10)

EOQ = √250,000

EOQ = 500 units

Answer:

Four Assumptions of EOQ Model:

1. Constant Demand: Demand for the product is known and remains constant throughout the year.
2. Instantaneous Replenishment: When an order is placed, the entire quantity is received at once with no delivery delays.
3. No Stockouts: Sufficient inventory is maintained, and stockout situations do not occur.
4. Constant Costs: Both ordering costs and carrying costs remain constant regardless of order quantity or time period.