Develop mathematical functions that compute the annual ordering cost and annual holding costs based on average inventory held throughout the year in order to arrive at a model for total cost.
To develop a model for total cost, we need to formulate mathematical functions for annual ordering costs and annual holding costs based on the average inventory held throughout the year.
1. Annual Ordering Cost:
The annual ordering cost refers to the cost incurred each time an order is placed for new inventory. It includes expenses such as order processing, transportation, and paperwork. The formula to calculate the annual ordering cost can be given as:
Annual Ordering Cost = Demand (D) * Ordering Cost per Order (S)
Where:
- Demand (D) is the total quantity of inventory demanded in a year.
- Ordering Cost per Order (S) is the cost incurred for placing a single order.
2. Annual Holding Cost:
The annual holding cost represents the expenses associated with holding or storing inventory over a one-year period. This includes costs such as warehousing, insurance, taxes, depreciation, and opportunity cost. The formula to calculate the annual holding cost can be expressed as:
Annual Holding Cost = Average Inventory (Q/2) * Holding Cost per Unit (H)
Where:
- Average Inventory (Q/2) is half of the total inventory quantity held on average throughout the year.
- Holding Cost per Unit (H) is the cost to hold one unit of inventory for a year.
3. Total Cost Model:
To calculate the total cost, we need to sum the annual ordering cost and the annual holding cost. The formula for the total cost (TC) can be represented as:
Total Cost (TC) = Annual Ordering Cost + Annual Holding Cost
Substituting the formulas we derived earlier, we get:
TC = D * S + (Q/2) * H, where
- TC is the total cost.
- D is the demand or total quantity of inventory demanded in a year.
- S is the ordering cost per order.
- Q is the total inventory quantity held on average throughout the year.
- H is the holding cost per unit.
By using this model, you can estimate the total cost based on the given demand, ordering cost, and holding cost.
To develop a mathematical model for total cost based on annual ordering and holding costs, we need to determine the functions that compute these costs.
1. Annual Ordering Cost:
The annual ordering cost is the cost associated with placing orders for additional inventory throughout the year. It is influenced by factors such as ordering frequency and cost per order. The mathematical function, AOC(Q), to compute the annual ordering cost based on the order quantity (Q), can be represented as:
AOC(Q) = D/Q * C
where:
- D is the annual demand for the product
- Q is the order quantity
- C is the cost per order
To compute the optimal order quantity that minimizes the total annual ordering cost, you can use various techniques like Economic Order Quantity (EOQ) or Total Cost approach.
2. Annual Holding Cost:
The annual holding cost is the cost associated with holding inventory throughout the year, including expenses like storage, insurance, and capital tied up in inventory. It depends on the average inventory held during the year and the cost per unit per year. The mathematical function, AHC(I), to compute the annual holding cost based on the average inventory (I), can be represented as:
AHC(I) = I * H
where:
- I is the average inventory level throughout the year
- H is the holding cost per unit per year
To compute the average inventory level (I), you can use various techniques, such as assuming a constant inventory level, or by considering the reorder point and lead time.
3. Total Cost Model:
The total cost model combines the annual ordering cost (AOC) and the annual holding cost (AHC) to calculate the overall cost of inventory management. The mathematical function, TC(Q, I), to compute the total cost based on the order quantity (Q) and the average inventory (I), can be represented as:
TC(Q, I) = AOC(Q) + AHC(I)
To find the optimal values of Q and I that minimize the total cost, you need to analyze the relationship between these factors and use optimization techniques like EOQ or other cost minimization approaches.
By developing and analyzing these functions, you can create a mathematical model to estimate the total cost of inventory management based on annual ordering and holding costs.