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Basic Inventory Models (Economic Order Quantity / EOQ Models)

Economic Order Quantity (EOQ) models are the most basic models of inventory management. The approach in EOW models is essentially to trade-off various relevant costs and derive an order quantity and time for placing an order such, that the total costs are minimized. This note discusses the basic EOQ model and the sensitivity of costs in EOQ model to various parameters. Later an extension of basic EOQ model is discussed in which case back-orders or shortage are allowed.

     1.1        Functions of inventory
Though inventory is an idle resource, it is almost essential to keep some inventory in order to promote smooth and efficient running of business.                                                                       


Consider the case – an enterprise that does not have any inventory. Clearly, as soon as the enterprise receives a sales order, it will have to order for raw materials to complete the order. This will keep the customers waiting. It is quite possible that sales may be lost. Also the enterprise may have to pay high price for some other reasons. On the other hand, inventory may promote sales by reducing customers waiting time. It is essential  to maintain the inventories in order to enhance stability of production and employment levels. Consider the case of seasonal items. Any fluctuation in demand can be met if possible, by either changing that part of production or with

inventories. However, if the fluctuation .... is not by changing the rate of production, one has to take into account the following cost.

Cost of increasing production and employment level:

   Employment and training
 Additional staff and service activities
(iii) Added shifts
 Overtime costs

Cost of decreasing production and employment level:

   Employee compensation
  Other employee costs
Staff, clerical and service activities
Total time costs

In other words, the use of seasonal inventories can often ... after balance of these costs. Broadly, some other functions of inventories are to :
(i)  protect against unpredictable variations (fluctuations) in demand and supply
  take the advantage of price discounts by bulk purchases
(iii) take the advantage of batches and longer production run.
(iv) provide flexibility to allow changes in production plans in view of change in demands, etc. and
(v)  facilitate intermittent production.

       Elementary Inventory Models (with Deterministic Demand)
Let us consider the inventory models in which demand is assumed to be fixed and completely pre-determined.

D - annual demand rate
V - unit purchase cost or unit cost of production (Rs./unit)
A - ordering or set up cost (Rs/order)
R - holding cost per Rs. per year (*Rs./Rs/year) (Inventory carrying charges factor)
B - shortage cost per Rs. short unit time (Rs/ Rs/ year)
Q - order quantity (to be determined)                                   See Inventory numericals here

       Inventory Cost
The heart of inventory analysis resides in the identification of relevant costs. Some of the important costs that apply to inventory situation are :

      Ordering or set up costs
These are costs associated with ordering or manufacturing goods through purchasing or manufacturing and are known as set up costs or cost of ordering. Set up costs are generally assumed to be independent of the quantity ordered or produced.

      Purchase cost or production cost
When large production runs are in process, these results in reduction of production cost per unit. Often, discounts are offered for the purchase of large quantities. In other words, often the unit cost of an item depends on the quantity procured or produced.

      Inventory Holding Cost
The cost associated with carrying or holding the goods in stock are known as carrying or holding costs. These costs arise due to the storage costs, property taxes on the items in inventory, interest on the invested capital (interest on value of the inventory items, spillage of the inventory items, depreciation of the inventory items, transportation and handling of the items in inventory, etc.

      Shortage or stock out costs
The costs that are incurred as a result of running out of stock are known as stock PUT TO SHORTAGE COSTS. As a result of shortages, sales or goodwill


may be lost. If the unfulfilled demand for the items can be satisfied at a later date (back order case), in this case cost of back orders are assumed to vary directly with the shortage quantity (in rupee value) and the delaying time (Rs./ Rs.) .
However, if the unfulfilled demand is lost (lost-sales case), in this case cost of shortages are assumed to vary directly with the shortage quantity (Rs./ unit shortage).


      Inventory Models

1. Economic Lot size or Economic Order Quantity Model

Assumptions :
1) The rate of demand for the item is deterministic and is a constant D units per per annum, independent of time.
2) Production rate is infinite, i.e production is instantaneous
3) Shortages are not allowed
4) Lead time is zero or constant independent of demand and the quantity ordered.
5) The entire quantity is delivered as a single package (or production in a single run).

Objective :
To minimise the average annual variable cost.

Problem :
To determine when an order should be placed and how much quantity should be ordered.
The annual variable costs for this problem are two types - 1) ordering or set-up cost and 2) Inventory holding cost.
As Q is the order quantity and D is the annual demand, the number of orders per years will be D/Q. Therefore, the annual ordering cost will be = A.D/Q ........ (1)

        Elementary Inventory Models (with Deterministic Demand)

Let us onsider the inventory models in which demand is assumed to be fixed and completely pre-determined. The inventory models with uncertain demand are considered later.

and leadSince quantity Q is ordered every time an order is placed and since the rate of demand is D, the time between two successive orders (T) will be 0/0. teh figure 1 shows the inventory level Vs Time rotationship.







From Fig.1 it is obvious that since the inventory is consumed at uniform rate and since maximum inventory level is Q, the averaghe inventory will be Q/2.
Hence, Average Investment in Inventory will be = Q/2.v
And the average Inventory Holding Cost will be = Q/2. ve .... (2)
Hence the total annual variable cost (TC) = ordering cost + Inventory Holding Cost.



The relationship between order quantity and
   Annual ordering cost
   Inventory Holding cost
   Total Annual Cost 



                                 Is given in Figure 2                                               Total cost = TC



                                                                                                   Average Inventory Holding

                                                                                                      Cost = Q/2. vr

                                                TC *


              0                               Q *

Economic Order Quantity and Reorder Level with Fixed Lead Time

In the above discussion and in figure 1 are considered that load time is zero. However, if lead time is constant the above results can be used without any modification.

If lead time is say constant and equal to L (in years) then during lead time, consumption is LD units. This élans that as soon as the inventory Q level reaches LO units, a new order will have to be released for Quantity G*, the new order will arrive exactly  after time period L at which time inventory level will be zero and the system will repeat itself. The inventory level at which the order is released is known as reorder level (R=0) i.e. Rp=LD. (Figure 3 shows the inventory level vs time the constant lead time situation.                         See Inventory numericals here










        4.3        Properties of EOQ model and Sensitivity Analysis
In the above nodal, various parameters are used such as demand (D), inventory carrying charges factor (r), ordering or set-up cost (A). These parameters are estimated and though they are assumed to be known, in real life what we have is an estimated value which may be different than real value for various reasons.


 For this reason it is important for practical purposes to test the results of the EQQ model and find how sensitive the results are to the changes in various parameters.
The sensitivity can be explored in various ways. Let the “true” rate of demand is D, “ true” value of order cost is A, “ true” value of inventory holding cost is r and “true” value 9of unit purchase cost or production cost is v. Then the “true” optimal value of order quantity (Q*) will be, (be eqn. 8),








          In case when shortage costs are infinitely high (i.e. when shortages are not allowed) b = ¥ and eqn. 17 will reduce to  

Which is same as equation (6), i.e. optimal order quantity for the E0-model. This is obvious since the EOQ model assume that no shortages are allowed which imply that the shortage are infinite.
See Inventory numericals here







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