__Costs Associated with Inventory:__

**1. Purchase (or production) cost:** The cost of producing one unit of a thing is its value. As soon as the price discounts are used, this cost increases significantly. The price is given as Rs. per unit.

**2. Capital cost:** Capital invested in an item (capital cost) is money that can’t be used to make other purchases. If the money were invested elsewhere, a return on the investment is expected. For this unreceived return, inventory expenses are charged. The amount of the charge reflects the percentage return expected from other investments.

**3. Ordering cost:** It is also known as **procurement cost**, **replenishment cost**, or **acquisition cost**. The cost of an order is the amount of money expended to get an item into inventory. This includes all expenses incurred from calling for quotes to the time the items are added to stock.

**Fixed costs** and **Variable costs** are the two categories into which costs can be separated.

**Fixed costs **are independent of the volume of orders, but variable costs vary according to the volume of orders. The salaries and wages of permanent employees involved in the purchase function and control of inventory, purchasing, incoming inspection, and accounting for purchase orders constitute the major part of the fixed costs. Order placement fees differ from one business to the next.

They are typically grouped under the following categories:

**(i) Purchasing:** The price of requisitioning materials, placing orders, following up, receiving, and analysing quotes, as well as the clerical and administrative costs related to purchasing.

**(ii) Inspection:** The cost of checking materials after they are received by the supplier for quantity and quality and maintaining records of the receipts.

**(iii) Accounting:** The price of processing payments, keeping track of purchases, and comparing supply to each request.

**(iv) Transportation costs:** The cost of transportation of the materials between different channel intermediaries.

**4. Inventory carrying cost (holding costs):** These are the costs associated with having a given level of inventory on hand and these costs vary in direct proportion to the amount of holding and period of having the stock in stores. The holding costs include:

(i) Storage costs (rent, heating, lighting, etc.)

(ii) Handling costs: Costs associated with moving the items such as the cost of labor, and equipment for handling.

(iii) Depreciation, taxes, and insurance.

(iv) Costs on record keeping.

(v) Product deterioration and obsolescence.

(vi) Damage from spoilage, breakage, theft, and loss because of its perishable nature.

**5. Shortage cost:** We incur a shortage cost or expense related to stock out when there is a demand for the product but the needed item is not in stock. The shortage costs include:

(i) Backorder costs.

(ii) Loss of future sales.

(iii) Loss of customer goodwill.

(iv) Ordering in a hurry and in small quantities come at an additional fee.

(v) contribution of lost sales revenue to profit loss.

The unsatisfied demand can be satisfied at a later stage (by means of backorders) or unfulfilled demand is lost completely (no back orders, the shortage costs become proportional to only the shortage quantity).

**Economic Order Quantity with Immediately Replenished Stock (Basic Inventory Model)**

**Assumptions**

a. Demand is predictable, constant, and deterministic.

b. Instant stock replenishment (lead time to zero)

c. Materials are fixed in price; bulk reductions are not permitted.

d. Quantity of orders has no effect on the cost of ordering.

A graphical representation of the model is shown in the following figure.

Let D represent the yearly demand (in units).

C_{0} = Ordering cost (Rs./order)

C_{h}= Inventory carrying costs (Rs./unit/unit time)

C_{p} = Price per unit

Q = Order Quantity

Q* = Economic order quantity

N = Annual volume of orders placed

T_{c} = Total cost per annum.

$$Annual\;ordering\;\cos t\;=\;No.\;of\;orders\;\times\;\frac{Ordering\;\cos t}{order}$$

$$Annual\;ordering\;\cos t\;=\;\left[\frac{Annual\;Demand}{Order\;quantity}\right]\;\times\;\left[\frac{Ordering\;\cos t}{order}\right]$$

$$Annual\;ordering\;\cos t\;=\;\frac DQ\;\times\;Co\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;….(1)$$

$$Annual\;inventory\;carrying\;\cos t\;=\;Average\;Inventory\;investment\;\times;inventory\;carrying\;\cos t$$

$$Annual\;inventory\;carrying\;\cos t\;=\;\left[\frac{Max.Inventory\;-\;Min.Inventory}2\right]\;\times;inventory\;carrying\;\cos t$$

$$Annual\;inventory\;carrying\;\cos t\;=\;\left(\frac{Q-0}2\right)\;\times;C_h\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;…..(2)$$

$$Annual\;total\;\cos t\;(T_c)\;=\;Annual\;ordering\;\cos t\;+\;Annual\;inventory\;\cos t$$

$$Annual\;total\;\cos t\;(T_c)\;=\;\frac{DC_0}Q\;+\;\frac{QC_h}2\;\;\;\;\;\;\;\;\;\;\;\;….(3)$$

To determine Economic order quantity (EOQ), differentiation, and Annual Total cost. Calculate the first derivative of Equation (3) with respect to Q by setting it to zero.

$$\frac{dT_c}{dQ}\;=\;\frac{-D.C_0}{Q^2}\;+\;\frac{C_h}2\;=\;0$$

$$Q^\ast\;=\;\sqrt{\frac{2DC_0}{C_h}}\;=\;0\;\;\;\;\;\;\;\;\;\;\;\;….(4)$$

$$Q^\ast\;is\;economic\;order\;quantity.$$

Note: If the inventory carrying cost is expressed as a percentage of the annual average inventory investment, then

$$Q^\ast\;=\;\sqrt{\frac{2DC_0}{C_PI}}\;=\;0\;\;\;\;\;\;\;\;\;\;\;\;\dots.(4a)$$

Where, I is expressed as a percentage of annual inventory investment. The ideal annual volume of orders placed

$$N^\ast\;=\;\frac{Annual\;demand}{Economic\;order\;quantity}\;=\;\frac D{Q^\ast}\;\;\;\;\;\;\;\;\;\;\;\;\dots.(5)$$

The optimal time interval between two orders.

$$T^\ast\;=\;\frac{Number\;of\;working\;days\;in\;a\;year}{T^\ast\;}\;\;\;\;\;\;\;\;\;\;\;\;\;\dots.(6)$$

The minimum total yearly inventory cost is given by

$$T_{cm}\;=\;\frac{DC_0}{Q^\ast}\;+\;\frac{Q^\ast C_h}2$$

$$Substituting\;for\;Q^\ast\;from\;equation\;(4)$$

$$T_{cm}\;=\;\frac{\displaystyle D.C_0}{\displaystyle\sqrt{\frac{\displaystyle2DC_0}{\displaystyle C_h}}}\;+\;\sqrt{\frac{\displaystyle2DC_0}{\displaystyle C_h}}\times\frac{C_h}2$$

$$T_{cm}\;=\;\sqrt{2DC_0.C_h}\;\;\;\;\;\;\;\;\;\;\;\;\dots.(7)$$