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INVENTORY AND CONTROL

Inventory is the physical stock of items held in any business for the purpose of future production or sales. In a production shop the inventory may be in the form of raw materials. When the items

are in production process, we have the inventory as in-process inventory and at the end of the production cycle inventory is in the form of finished goods. We shall be dealing only with the

finished goods inventory. The problem of determining inventory policies is not a new concept  beginning. It is only in he last two decades that it has been tackled with quantitative techniques

and mathematical models, a method amenable to optimization.

Inventory planning is the determination of the type and quantity of inventory items that would be required at future points for maintaining production schedules. Inventory planning is generally based on information from the past and also on factors that would arise in future. Once this sort of planning is over, the control process starts, which means that actual and planned inventory positions are compared and necessary action taken so that the business process can function efficiently.

In inventory control, we are primarily concerned with the inventory cost control. The aim is focussed to bring down the total inventory cost per annum as much as possible. Two important questions are

(1) how much to stock or how much to buy and

(2) how often to buy or when to buy.

An answer to the above questions is usually given by certain mathematical models, popularly known as ‘economic order quantity models’ or ‘economic lot/batch size models (E.O.Q.).’

INVENTORY COSTS

There are four major elements of inventory costs that should be taken for analysis, such as

(1)Item cost, Rs. C1/item.

(2)Ordering cost, Rs. C2/order.

(3)Holding cost Rs. C3/item/unit time.

(4)Shortage cost Rs. C4/item/Unit time.

Item Cost (C1)

This is the cost of the item whether it is manufactured or purchased. If it is manufactured, it includes such items as direct material and labour, indirect materials and labour and overhead expenses. When the item is purchased, the item cost is the purchase price of 1 unit. Let it be denoted by Rs. C1 per item.

Purchasing or Setup or Acquisition or Ordering Cost (C2)

Administrative and clerical costs are involved in processing a purchase order, expediting, follow up etc., It includes transportation costs also. When a unit is manufactured, the unit set up cost includes the cost of labour and materials used in the set up and set up testing and training costs. This is denoted by Rs. C2 per set up or per order.

Inventory holding cost (C3)

If the item is held in stock, the cost involved is the item carrying or holding cost. Some of the costs included in the unit holding cost are

(1)Taxes on inventories,

(2)Insurance costs for inflammable and explosive items,

(3)Obsolescence,

(4)Deterioration of quality, theft, spillage and damage to times,

(5)Cost of maintaining inventory records.

This cost is denoted by Rs. C3/item/unit time. The unit of time may be days, months, weeks or years.

Shortage Cost (C4)

The shortage cost is due to the delay in satisfying demand (due to wrong planning); but the demand is eventually satisfied after a period of time. Shortage cost is not considered as the opportunity cost or cost of lost sales. The unit shortage cost includes such items as,

(1)Overtime requirements due to shortage,

(2)Clerical and administrative expenses.

(3)Cost of expediting.

(4)Loss of goodwill of customers due to delay.

(5)Special handling or packaging costs.

(6)Lost production time.

This cost is denoted by Rs. C4 per item per unit time of shortage.

INVENTORY MODELS (E.O.Q. MODELS)

The inventory control model can be broadly classified into two categories:

(1)Deterministic inventory problems.

(2)Probabilistic or stochastic inventory problems.

In the deterministic type of inventory control, the parameters like demand, ordering quantity cost, etc are already known or have been ascertained and there is no uncertainty. In the stochastic inventory control, the uncertain aspects are taken into account.

First let us consider the inventory control of the deterministic type. There are four EOQ models which are discussed below. The first one is the well-known Wilson’s inventory model.

Model 1: Purchasing model with no shortages: (Wilson’s model)

The following assumptions are made in deriving the formula for economic order quantity.

(1)Demand (D) is at a constant rate.

(2)Replacement of items is instantaneous (lead time is zero).

(3)The cost coefficients C1, C2, and C3 are constant.

(4)There is no shortage cost or C4 = 0.

This mode represented graphically in fig. 1. This is also known as a saw tooth model (because of its shape).

Quantity Q =Im

Fig. 1

Fig. 2

In this model, at time t = 0, we order a quantity Q which is stored as maximum inventory, 1m. The time ‘t’ denotes the time of one period or it is the time between orders or it is the cycle time. During this time, the items are depleting and reaching a zero value at the end of time t. At time t another order of the same quantity is to be placed to bring the stock upto Q again and the cycle is repeated. Hence this is a fixed order quantity model.

The total cost for this model for one cycle is made up of three cost components.

(1)The economic order quantity.

(2)The time between orders.

(3)The number of orders per year.

(4)The optimum annual cost if the cost of item is Rs. 2 per item.

Solution:

Note that the holding cost is given per month and convert the same into cost per

year.

C1 = Rs. 2/item C2 = Rs. 100/order C3 = Rs. 0.80/item/month

In this model, shortages are allowed and consequently a shortage cost is incurred. Let the shortages be denoted by ‘S’ for every cycle and shortage cost by C4 per item per unit time. This model is illustrated in Fig .3

Fig. 3

Fig. 3 shows that the back ordering is possible (i.e.) once an order is received, any shortages can be made up as the items are received. Consequently shortage costs are due to being short of stock for a period of time. The cost per period includes four cost components. Total cost per period = Item cost + Order cost + Holding cost + Shortage cost Item cost per period = (item cost) × (number of items/period)

Example:

The demand for an item is 18000 units/year. The cost of one purchase is Rs. 400. The holding cost is Rs.

1.2 per unit per year. The item cost is Rs. 1 per item. The shortage cost is Rs. 5 per unit per year. Determine:

(a)The optimum order quantity.

(b)The time between orders.

(c)The number of orders per year.

(d)The optimum shortages.

(e)The maximum inventory.

(f)The time of items being held.

(g)The optimum annual cost.

Model 3:

Manufacturing model with no shortages

In this model the following assumptions are made:

(1)Demand is at a constant rate (D).

(2)All cost coefficients (C1, C2, C3) are constants.

(3)There is no shortage cost, or C4 = 0.

(4)The replacement rate is finite and greater than the demand rate. This is also called replenishment rate or manufacturing rate, denoted by R.

Schematically, this model is illustrated in fig 4

The total cost of inventory per period is the sum of three components: item cost, order cost and items holding cost.

Let Im be the maximum inventory, t1 be the time of manufacture and t2 be the time during which there is no supply.

In this model, all items required for a cycle are not stored at the beginning as in Wilson’s Model. The items are manufactured at a higher rate than the demand so that the difference (R–D) is the existing inventory till the items are exhausted.

Item cost/period = C1Q (32) Order cost/period = C2 (33)

(a)The optimum manufacturing quantity.

(b)The maximum inventory.

(c)The time between orders.

(d)The number of orders/year.

(e)The time of manufacture.

(f)The optimum annual cost if the cost of the item per unit is Rs. 2.

Model 4.

Manufacturing model with shortages

The assumptions in this model are the same as in model 3 except that the shortages are also considered. This model is illustrated in the figure in power point presentation of the lecture, given on the next page.

There are four components of inventory costs in this model. They are

(1) Item cost.

(2)Set up or order cost.

(3)Items holding cost.

(4) Shortage cost. The items t1, t2, t3 and t4 are as represented in figure on the next page. The total cost per period t is C′= CQ + C + C (t + t )I 2 + C (t +t ) S 2 (46) 1 2 312^m

The values of t1, t2, t3 and t4 are to be determined in terms of Q and S, so that the cost is expressed as a function of only two variable Q and S.

ORDER QUANTITY WITH PRICE-BREAK

The concept of Economic Order Quantity fails in certain cases where there is a discount offered when purchases are made in large quantities. Certain manufacturers offer reduced rate for items when a larger quantity is ordered. It may appear that the inventory holding cost may increase if large quantities of items are ordered. But if the discount offered is so attractive that it even outweighs the holding cost, the probably the order at levels other than the EOQ would be economical. An illustration is given in the following example and the rationale is explained.

Example: A company uses 12000 items per year supplied ordinarily at a price of Rs. 3.00 per item. Carrying costs are 16% of the value of the average inventory and the ordering costs are Rs. 20 per order. The supplier however

The EOQ at Rs. 2.90 per item = 1017. But the price per item is Rs. 2.9 only if the items are ordered in the range of 2000 to 3999. This is therefore an infeasible solution. Similarly the EOQ at Rs. 2.85 per item is 1026. This price is valid only for items ordered in the range 4000 or more. This is also an infeasible solution.

We follow the routine procedure and calculate the cost for various order sizes: 1000, 2000, 4000,

Order size

1. Assume the following price structure

Purchase cost per order = Rs. 25 Cost of the item = Rs. 10 Annual demand = 950 Units Carrying cost = Rs. 2/Unit/year

2. Find the optimal order quantity for a product for which the price-breaks are as follows:

The monthly demand for the product is 600 units. The storage cost is 15% of unit cost and the cost of ordering is Rs. 30 per order.

DYNAMIC ORDER QUANTITY

The basic assumption in the derivation of Economic Order Quantity models discussed previously is that the demand is uniform. But in certain situations the demand is not uniform. It may rise and fall, depending on seasonal influences. A general method is discussed below that can be applied to any pattern of varying demand due to seasonal or irregular variations.

Consider the following example to illustrate how a varying demand problem can be tackled. This is known as Dynamic Order Quantity model.

Example: The requirements for 12 months are given below:

Solution:

To calculate the dynamic order quantity we can adopt the following procedure.

The first month’s requirement has to be ordered in the first month itself at a procurement cost of Rs. 20. Now we have to decide whether the second month’s requirement can also be ordered along with first month’s requirement. This involves additional carrying cost, but this will result in saving an extra set up. Hence if the saving on set up costs outweighs the carrying costs, then we include the second month’s requirements along with the first month. Similarly a decision can be taken whether to include the third month’s requirement in the first month itself. This procedure is followed until the procurement costs and carrying costs are balanced.

Let n represent the month, n = 1, 2, …, 12. Let Rn be the requirement during nth month and Rn+1 be the requirement during (n + 1)th month. If the (n + 1)th month requirement namely Rn+1 is absorbed in nth month itself, the additional procurement cost is saved.

Hence the saving in procurement cost is (C2/n). But the additional carrying cost is C3 n (Rn+1). If the additional carrying cost is less than the procurement cost, then the (n+1)th month requirement is to be ordered also with nth month.

If the answer to the above inequality is yes, then, the future month’s requirements are included in the first month itself. If the answer is ‘no’, then that requirement is to be ordered afresh and this is treated as month n = 1. In this example C2/C3 = 20/0.01 = 200. All the eleven information can be represented in the table as shown.

Hence we order three times a year in the first month, seventh month and tenth month, the batch sizes being 92, 110 and 70 respectively.

Exercise:

Compute the dyanamic EOQ is for the following requirements.

ABC ANALYSIS

The ABC analysis is the analysis that attracts management on those items where the greatest savings can be expected. This is a simple but powerful tool of statistical sampling in the area of inventory control or materials management.

In this analysis, the items are classified or categorized into three classes, A, B and C by their usage value. The usage value is defined as,

The ABC concept is based on Pareto’s law that few high usage value items constitute a major part of th capital invested in inventories whereas a large number of items having low usage value constitute an insignificant part of the capital. It too much inventory is kept, the ABC analysis can be performed on a sample. After obtaining the random sample the following steps are carried out for the ABC analysis.

STEP 1:

Compute the annual usage value for every item in the sample by multiplying the annual requirements by the cost per unit.

STEP 2:

Arrange the items in decending order of the usage value calculated above.

STEP 3:

Make a cumulative total of the number of items and the usage value.

STEP 4:

Convert the cumulative total of number of items and usage values into a percentage of their grand totals.

STEP 5:

Draw a graph connecting cumulative % items and cumulative % usage value. The graph is divided approximately into three segments, where the curve sharply changes its shape. This indicates the three segments A, B and C.

The class A items whose usage values are higher are to be carefully watched and are under the strict and continued scrutiny of the senior inventory control staff. These items should be issued only an indents sanctioned by the staff. The class C items on the other extreme can be placed on the shop floor and the personnel can help themselves without placing a formal requisition. The class B items fall in between A and C.

ABC concept conforms to the consideration implied in the EOQ model. ‘A’ items have high inventory carrying costs and should therefore be placed with EOQ concept. The ‘C’ items require very little capital and have therefore low inventory carrying costs. Hence, they can be purchased in bigger lots. ‘B’ items are usually placed under statistical stock control.

Example:

Perform ABC analysis on the following sample of 10 items from an inventory.

Solution:

STEP 1:

Calculate usage value by multiplying annual usage of each item with its unit cost and tabulate them and assign rank by giving rank to the largest usage value as in table below.

STEP 2:

Arrange items as per ranking and calculate cumulative usage and % cumulative value as in the table below.

If you draw the figure, you will see that the curve changes at points (say) X and Y. The items upto X is classified as class A items and between X and Y as class B and the rest as class C items.

SOME DEFINITIONS

Lead-time:

This is defined as the time interval between the placing of the orders and the actual receipt of goods.

Lead-time Demand:

This is the lead-time multiplied by demand rate. For example, if the lead-time is 3 weeks and the demand is at the rate of 50 items per week, then the lead-time demand is 3 x 50 = 150 items.

The lead-time may not be constant. For one batch, a vendor may take 45 days and for the next batch 50 days and so on. Lead-time itself is therefore a stochastic variable. This complicates the problem of accumulating stock over the period encompassed by the lead-time. Lead-time may also be forecast exponentially as is done with the demand.

Safety stock or Buffer stock:

Lead-time demand is the stock level, which, on the average is sufficient to satisfy the customer’s orders as the stocks are being replenished. “On the average” would mean that during this period of replenishment 50% of the customer’s order can be filled and the remaining 50% may either be refused or back ordered to be filled later. The reason for this is obvious. Forecast is after all a point estimate only. If the demand is greater than the forecasts, the customers would not be serviced. If the demand is less then the forecasts, overstocking would occur. When these two variables, stock level and service to the customer, are summed up over thousands of stock level and service to the customer, are becomes a major problem for an organisation to find an acceptable compromise between the two. Sometimes the management in an organisation would like to limit the disservice to the customer down to 5% or 10% at the cost of extra stocking. This extra stock in excess of the lead-time demand is called the safety stock. Saftely stock may be expressed as percentages of the lead-time demand. It may be computed in different ways.

Reorder level:

This is defined as the level of the inventory at which the order is placed. It has generally two components (i) Lead time Demand and (ii) Safety Stock.

Reorder level (ROL) = Lead time demand (LTD) + Safety stock (SS).

Computation of Safety Stock:

As discussed earlier, if the demand exceeds the forecast, the result is bad service to the customer and if the demand is less than the forecast figure, this results in overstocking. Thus there is a forecast error. This error is assumed to be normally distributed, with zero mean. If the standard deviation of the forecast error is calculated, then safety stock may be set with the desired confidence level to result in not more the 5 or 10% shortages etc.

Example:

A company uses annually 50000 units of raw materials at a cost Rs. 1.2 per item. Ordering cost of items is Rs. 45 per order and item carrying cost is 15% per year of the average inventory.

1. Find the economic quantity.
2. Suppose that the company follows the EOQ policy and it operates for 300 days a year, that the procurement time is 12 days and mum, minimum and average inventories.

Example:

A scrutiny of past records gives the following distributions for lead time and daily demand during lead time.

Lead time distribution

Demand distribution

What should be the buffer stock?

Solution:

Computation of average or mean lead time.

Average lead-time demand = Average lead time x Average demand/day = 6 x 3 = 18.

Maximum lead-time demand. = Max. lead time x Max. demand/day = 10 x 7 = 70

∴ Buffer stock = Max. lead time demand – Average lead time demand

*Example 7.7.3

For a fixed order quantity system find the EOQ, SS, ROL and average inventory for an item with the following data. Demand = 10000 units Cost of item = Re. 1 Order cost = Rs. 12

Holding cost = 24% Post lead times = 13 days

Example: An airline has determined that 10 spare brake cylinders will give them stock out risk of 30%, whereas 14 will reduce the risk to 15% and 16 to 10%. It takes 3 months to receive items from supplier and the airline has an average of 4 cylinders per month. At what stock level should they reorder assuming that they wish to maintain an 85% service level.

Solution:

Lead time demand = 3 x 4 = 12 items

Safety stock at 85% service = 15% disservice or 15% stock out risk = 14 items

Reorder level = 12 + 14 = 26 items.

Example: Data on the distribution of lead time for a motor component were collected as shown. Management would like to set safety stock levels that will limit the stock out to 10%.

How many weeks of safety stock are required to provide the desired service level?

Solution:

Average lead time = ∑ (lead time x frequency)/ ∑ frequency = (10 + 40 + 210 + 160 + 150 + 60 + 70 + 80)/200 = 770/200 = 3.85 weeks

Upto 90% service level max. lead time is 6 weeks. Hence 6 – 3.85 = 2.15 weeks of stock would provide the service level of 90% (Note: If the lead time is given as a continuous time distribution take the mid point.)

Example:

Demand for a product during an order period is assumed to be normally distributed with mean of 1000

units and standard deviation of 40 units. What % service can a company expect to provide (i) if it satisfies the average demand only (ii) if it carries a safety stock of 60 units.

Solution:

1. If the company provides only average demand, we can expect only 50% service level.
2. The standard normal variate Z is computed with the following formula. Z = (Safety stock – 0)/ Standard deviation

(60 – 0)/40 = 1.5 The area under normal curve for Z = 1.5 is 0.4332. ∴ Service level = 0.50 + 0.4332 = 0.9332 04 93.32%

Example:

A manufacturer of water filters purchases components in EOQ’s of 850 units/order. Total demand averages 12000 components per year and MAD = 32 units per month. If the manufacturer carries a safety stock of 80 units, what service level does the this give the firm?

Solution:

Standard deviation = _Ï€ ² × MAD = MAD /0.8 = 32 /0.8 = 40

Z = (S S – 0)/S.D = (80 – 0)/40 = 2 The area under normal curve for Z = 2 = 0.4772. Service level = 0.9772 or 97.72%

Example:

A firm has normally distributed forecast of usage with MAD = 60 units. It desires a service level, which

limits the stock, outs to one order cycle per year.

(1)How much safety stock should be carried if the order quantity is normally a week’s supply?

(2)How much safety stock should be carried if the order quantity is weeks supply.