Several models have been developed for the purpose of inventory planning and control. The basic purpose behind such modelling is to arrive at the level of optimum investment in inventories. As will be evident from the discussion that follows, these models allow one to figure out the optimum lot size, i.e. the number of units that should be ordered each time.
There exist basically two kinds of models: deterministic and stochastic or probabilistic. The deterministic models are built on the premise that there is no uncertainty associated with the demand and replenishment or lead times.
The probabilistic model take cognizance of the fact that there is always some uncertainty associated with the demand pattern and lead times.
For the purpose of exposition, we shall now proceed to develop a deterministic model for arriving at the Re-order Quantity or the Economic Order Quantity (EOQ). This is an important concept in the purchase of raw material and in the storage of finished goods and in-transit inventories. We shall determine optimal order quantity for a particular item of inventory. In this exercise, we are going to arrive at the optimal order quantity of an item of inventory, given its forecast usage, the ordering cost and the carrying cost. Ordering cost can mean purchase or production.
Let us assume that the usage of this particular item is known with certainty and that the usage is stationary or steady throughout the period of time being analyzed. In essence, what we are assuming is that if the usage is 5200 units a year, the usage every week is 100 units. Goods are used evenly throughout the year. It is noteworthy that the EOQ model can be modified to take account of increasing or decreasing use over time. For the purpose of this exercise, such modifications are not being considered. We are assuming that the cost per order or the ordering cost, k, is constant regardless of the size of the order. As discussed earlier, k, represent the clerical and administrative and other costs involved in placing an order for the purchase of raw materials. For finished goods inventories, the cost of ordering involves scheduling a production run and for in-transit inventories it involves basically record keeping. Obviously, the total ordering cost is the cost per order times the number of orders placed.
The average holding cost or carrying cost per unit represent the cost of inventory storage, handling, insurance, etc, and the required rate of return on the investment in inventories. We are assuming that these costs are constant per unit of inventory per unit of time. Therefore, the total carrying cost for a period is the average number of units of inventory for the period times the carrying cost per unit.
We are also assuming that inventory orders are filled without delay, since out-of-stock items can be filled without delay, there is no need to maintain a safety stock or buffer stock.
Since the usage has been assumed to be steady and there is no buffer stock, the average inventory can be expressed as Q/2, where Q = quantity per order and this quantity ordered is assumed to be constant over the period. Let us also assume that the particular item in question is purchased, the total cost involved on this count is the cost per unit times the number of units purchased.
Where KD/Q represents ordering cost PD represents cost of purchase of the item in question, and C Q/2 represents the holding cost or the carrying cost.
For an optimal solution, we need to minimize the total associated cost. We would therefore set the first derivative = O and find out whether the second derivative is positive.
DTC/dQ = -KDQ– 2/2 + C = O
therefore KD/Q2 = C/2 or CQ2 = 2KD or Q2 = 2 KD/C√2KD or Q * or the economic order quantity = ----- C
The EOQ model is useful in so far as it tells the amount to order and the best timing of our orders in the case of raw materials.
With respect to finished goods inventories, it enables us to exercise better control over the timing and size of production runs.
As a whole, this model provides a decision rule regarding the replenishment of inventories, the time and the amount to be replenished.
A graphical representation of the EOQ relationship appears in below Figure . In the figure, we plot ordering costs, carrying costs and total costs – the sum of the first two costs; we see that whereas carrying costs vary directly with the size of the order, ordering cost vary inversely with the size of the order. The total cost time declines at the first as the fixed cost of ordering are spread over more units. The total cost time begins to rise when the decrease in average ordering cost is more than offset by the additional carrying costs. Point x then represents the economic order quantity, which minimizes the total cost of inventory.
The EOQ formula taken up in this unit is a very useful tool for inventory control. In purchasing raw material or other items of inventory, it tells the amount to order and the best timing of our orders. For finished goods inventory, it enables us to exercise better control over the timing and size of production runs. In general, the EOQ model gives us a rule for deciding when to replenish inventories and the amount to replenish.
Illustration – the usage of an inventory item during the year is estimated at 2000 unit. The ordering cost works out to N100/order and the holding cost is estimated at N10 per unit per year. The cost of the item i.e. the purchase price is N1 per unit. By applying the EOQ model, we can directly arrive at the EOQ as follows:Q* =√2(100).(2000)/10 = 200 units.
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