INVENTORY MODELS

Introduction

An inventory is any stock of economic resources that is stored for future use. If a

commodity is ordered frequently, then the costs of ordering are high. On the other hand,

ordering more units less frequently saves on ordering costs but increases the expense of

keeping a larger inventory.

Thus the management problem in this case is: How frequently should materials be

ordered?

The purpose of inventory

The following are the major reasons for maintaining an inventory:-

1. Protection against fluctuating demand: Inventories are kept to meet peak demand.

E.g. blood is stored in hospitals in quantities sufficient to meet the needs of a accident

2. Protection against delayed supply – A strike by the suppliers’ employees is one

reason why deliveries may not arrive in time. Lack of materials at the supplier level,

strikes in the transportation network are other possible causes for shortages.

Inventories are kept as a buffer that can be used until late deliveries arrive.

3. Benefits of large quantities – purchasing large quantities of an item often entitles the

buyer to a price discount. Similarity in the case of manufacturing large production

lots, the utilization of more efficient (automated) equipment can be justified by the

reduction in the per unit manufacturing cost.

4. Primary basis for business – Retail operations involving customer examination and

selection require fully stocked shelves and complete inventory.

5. Savings on ordering costs – ordering in large quantities reduces the number of times

that an order must be placed and processed. Since a fixed cost is associated with

placing each order, the fewer times one places an order, the lower the total cost of

ordering will be.

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6. Other reasons – Inventories are kept for several other reasons. An inventory may

improve the bargaining power of a firm with a supplier (or with its own employees)

by making the company less vulnerable to delays or stoppages.

Inventory is also kept so that machines can be shut down for overhauls.

An inventory of labor is maintained to meet fluctuating production demands in order

to reduce hiring, firing and training costs.

The importance of Inventory Management

Proper inventory management may be one of the most important functions of

management. Excess inventories are costly to store but insufficient inventories may result

in loss of market share or idle employees.

The task of inventory control is a part of a management function called materials

management, which is concerned with the acquisition, distribution, storage, and disposal

of materials and parts in organizations.

The structure of the inventory system

We consider inventory models that pertain only to an individual item in stock. This

means, for example, that with an inventory system of three different items, the model

must be applied three times.

An inventory system involves a cyclical process that is assumed to run on several periods

and whose major characteristics are described next.

Inventory level – An item is stocked in a ware-house, store or any other storage area.

This stock constitutes an inventory. The size of the inventory is called inventory level or

the inventory on hand.

Demand and depletion – The inventory is depleted as demand occurs. Assume that one

starts with an inventory of 100 units. As time passes, the inventory level declines due to

the demand for the item in stock. The rate of demand determines the depletion rate and

the inventory level.

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The higher the rate of demand, the quicker the inventory is reduced. The rate of demand

can be constant e.g. 5 units a day or it may fluctuate (e.g. three units on the first day and

seven on the second). A constant demand reduces the inventory level in equal steps. The

steps of constant demand can be approximated by a straight line. A fluctuating or

variable demand is shown by unequal steps and can be approximated by a curve.

Re-ordering – To re-build an inventory, the item is replenished periodically. When the

inventory level is reduced to a certain level called the re-order point, a replenishment

order is placed. The time between reordering and receiving the shipment is called the

lead time.

Replenishment, shortages and surpluses – In some basic inventory models, it is

assumed that the re-ordering is scheduled so that the replenishment will arrive exactly

when the inventory level reaches zero.

However, if the demand fluctuates and/or the lead time varies, the shipment may arrive

either before or after the stock is completely depleted i.e. the depletion and replenishment

do not coincide. In such a case, a surplus or a shortage will occur.

If the shipment arrives after depletion, then the demand cannot be met and a shortage or

stock out will occur.

When the shipment arrives prior to depletion, an inventory level larger than zero or a

surplus exists. Shortages can be eliminated or reduced by establishing a buffer or safety

stock.

The Average Inventory – for purposes of inventory decision making as well as other

managerial uses such as insurance and taxation, the concept of an average inventory is

used. Assume that during a 5-day period the inventory levels are as follows.

M T W T F

16 12 8 4 0

Then the average inventory is

units

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If the demand is constant, the average inventory can be computed by adding the

inventory at the beginning of the cycle (16) to the inventory at the end of the cycle (0) (in

this case) and dividing it by 2.

If the inventory at the end of the cycle is zero (as in this example), the average inventory

equals one-half of the initial (or maximum) inventory.

If the demand is not constant, a more complicated formulation is required.

Inventory Problems & Decisions

The major problem of inventory management is determining the appropriate inventory

levels. This problem is related to the problem of “how much” to order since the amount

ordered determines, in part, the inventory level. Also related is the problem of “when” to

order. These three issues appear together in most models.

The most common criteria considered in inventory analysis are inventory-related costs.

Inventory models compute these costs, total them, and then search for a policy or an

alternative solution that minimizes the total cost.

Inventory costs

Inventory problems are usually examined from a cost rather than a profit stand point.

The major costs are:

Ordering cost (k) – Ordering cost includes all the expenses of placing orders. It is

assumed to be a fixed cost per order, i.e. each time an order placed, the same expenses

occur regardless of how many units are ordered.

Holding (or carrying cost (H) – The expenses of holding or carrying the inventory

include

i) Cost of capital – The interest paid on the capital invested in inventories or the

opportunity cost of doing something else with the money.

ii) Storage – costs of maintaining the storage space. This includes rental fees,

security services, lights (heat) etc.

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iii) Store keeping operations – Expenses such as record keeping and taking of

physical inventory.

iv) Insurance and taxes.

v) Obsolescence and deterioration costs of the items stored.

All holding (carrying) costs are totaled and expressed either in terms of dollars per item

per year, or in percentage of the value of the inventory.

E.g. a chair costs sh.40. To keep the chair in inventory for one year will cost H= sh.10.

Alternatively we can say that the holding cost is 25% of the value of the item.

Shortage (or stock out) costs (G)

Shortage costs occur when an item is out of stock and demand is unsatisfied. Depending

on the item under consideration, shortage costs may include of the following

i) In the case of raw materials: Costs of idled production, spoilage of

products/materials, and the cost of placing and fulfilling special (expediting)

orders.

ii) In the case of finished goods: Cost of ill will (loss of customers) due to

inability to deliver or due to late deliveries.

iii) In the case replacement parts: Cost of idle machines, idle labor, spoilage of

materials and delays in shipment.

iv) In other cases: The shortage of blood or ambulance may cost a life and

shortage of fire engines may result in excessive damage caused by a fire.

Shortages may be temporary (back orders), in which case they are eliminated when the

supply arrives; or permanent, in the sense that sales are lost.

Item cost (C) – Item (or unit) cost is the price paid for one unit of the commodity under

consideration. It is not a direct inventory cost, as the items must eventually be procured,

but it may be influenced by inventory decisions.

E.g. ordering large quantities may result in a lower price per unit due to quantity

discounts.

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Economic Order Quantity Model (EOQ)

The EOQ model is the most elementary of all inventory models. Its objective is to

determine the optimal quantity to order. It answers the following questions:

i) How much should be ordered each time?

ii) When should it be ordered?

iii) What will the total cost be?

iv) What will the average inventory level be?

v) What will the maximum inventory level be?

Assumptions of EOQ

The EOQ model assumes the following

i) The demand for the item is constant over time e.g. two units per day.

ii) Within the range of quantities to be ordered, the per unit holding cost and

ordering cost are independent of the quantity ordered.

iii) The replenishment is scheduled in such a way that shipments arrive exactly when

the inventory level reaches zero. Therefore there will never be a shortage or a

surplus.

iv) Since only one item is being considered, orders for different items are

independent on each other.

Example

A university uses 1200 boxes of typing paper each year. The university is trying to

determine how many boxes to order at one time. The information it considers is

Annual demand, D = 1200 boxes

Ordering cost, k = $5 per order

Holding cost, H = $ 1.20 per box, per year

The problem is to find the quantity to be ordered, Q.

We consider three possible ordering polices; annually, quarterly & monthly.

1. Annual policy: Order once a year: Therefore Q = 1200 boxes

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2. Quarterly: Order once a quarter, four times a year. boxes at a time.

3. Monthly: Order once a month, 12 times a year boxes at a time.

Other ordering policies could also be considered e.g. weekly, semi-annually, or once

every 2 years. The problem faced by management is: which ordering policy is the best?

Solution: Using a trial and error approach

We compute the total annual inventory cost for each of the suggested policies. The

policy with the lowest total cost is the best. The total cost is given by:

Total annual

Inventory cost

= Total annual

ordering cost

+ Total annual

Holding cost

(1)

TC = TO + TH

Step 1: Find total annual ordering cost, TO

The total annual ordering cost is given as the number of times an order is placed, N

multiplied by the ordering cost, k i.e. TO = N k (2)

But the number of times an order is placed during the year is given by the total yearly

demand, D, divided by the order quantity, Q

i.e. (3)

Thus the equation for TO is

(4)

TO in the 3 proposed policies is

Annual: N= 1, k = 5 TO = 1(5) = 5

Quarterly: N= 4, k = 5 TO = 4(5) = 20

Monthly: N= 12, k = 5 TO = 12(5) = 60

Notice that as the order quantity, Q, increases, the total annual cost of ordering decreases.

This is because the larger the order size, the fewer the number of orders placed per year.

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Step 2: Find total annual holding cost, TH

The total annual holding cost can be computed by multiplying the daily holding cost by

the number of units in inventory each day of the year, and summing. However, since the

inventory level is changing from day to day, the number of units in the inventory

fluctuates over time thus it is easier to solve this problem by using the average inventory

on hand over the year and multiplying by the yearly holding cost.

When the demand is constant, the average inventory is the midway point between the

highest and lowest inventory level.

Since one of the EOQ assumptions requires that the lowest inventory level be zero, the

average inventory equals exactly of the maximum inventory.

In the EOQ model, the maximum inventory equals the order quantity, Q. Consequently,

the average inventory = of Q

i.e. Average inventory (5)

Therefore, the total annual inventory holding cost, TH, will be

(6)

The total annual inventory holding cost for the three proposed ordering policies is

Annual : Q = 1200,

Quarterly : Q = 300,

Monthly : Q = 100,

Step 3: Compute Total Annual Inventory Cost, TC

Using equation (1) the total annual cost for the proposed policies is

Policy TO + TH = TC

Annual 5 + 720 = 725

Quarterly 20 + 180 = 200

Monthly 60 + 60 = 120

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Comparing the three alternatives, the best ordering policy is ‘monthly’ since it has the

lowest total cost of $120.

However, since other possible ordering policies (e.g. weekly, semi annual) were not

checked, no assurance exists that monthly ordering is indeed the optimal policy. To

check all possible polices may involve much computational work. Therefore a more

efficient method is provided through the EOQ formula.

The EOQ Formula

We have shown that the total cost, TC, can be expressed as

(7)

Where D is the annual demand, k is the ordering cost, H is the holding cost, and Q is the

quantity to be ordered.

The problem is to find that Q for which T.C is the minimum

The EOQ can be obtained through calculus

Setting the 1st derivative equal to O

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Thus, the optimal value of Q, i.e. the EOQ is

where K is the ordering cost (in dollars), D is the annual demand (in units) and H is the

holding cost (in dollars per unit per year)

NB: To verify that this is a minimum point, we check for positive 2nd derivative

Since the 2nd derivative is positive (K, D and Q can only take positive values), the point is a minimum point indeed

Solution to the example:

boxes.

The optimal solution calls for an order size of 100 boxes at a time. For a yearly demand

of 1200 this means 12 orders a year or one a month. Thus the monthly order policy is

indeed optimal.

Additional Information Provided by EOQ

In addition to the size of the order to be placed, the EOQ can be used to provide the

following information

i) The best number of orders to be placed in year. Using equation (3) we find

times

ii) The maximum inventory on hand and average inventory level. The max inventory is

equal to or 100 in this example. The average is one half of , which is 50 in

this example.

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iii) The number of days supplied. The computation of the EOQ also helps to provide the

number of days supply, d. This information tells management the length of each

inventory cycle. This is given as

In this example

days

NB: If the company works less than 365 days a year, use the actual number of days

worked.

iv) The total annual cost. Using equation (7), the total annual cost (excluding the cost of

goods themselves) can now be computed.

Notice that the two components of the total cost, the ordering cost and the holding cost

must equal each other whenever the optimal Q is used ($60 each in the example).

v) The monetary value of an optimal order and of the average inventory. This

information is important since the monetary value of the average inventory is useful

for such purposes as cash flow determination, tax assessment, and calculation of

depreciation.

The monetary value of the EOQ is obtained by multiply by the unit cost. Assume in

the example that the cost of one box of paper is $10. Thus Q the dollars is

.

That is, the university orders $1000 worth of supplies at a time. Similarly the monetary

value of the average inventory is

.

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Application of EOQ model

In applying the EOQ formula, the following points may be helpful.

Holding cost given as a percentage of the value

It is sometimes common to express the holding cost as a percentage of the value in

inventory. E.g. it may be stated that the inventory holding costs are 30% per year. This

means that if the item is worth $20, then H is per item, per year.

In general , where I = annual percentage and C = per item cost.

When the demand is given in Monetory terms.

In some cases the demand for an item is given in monetary value rather than in units.

Example: A recreation department’s annual budget for supplies is $200,000. The

ordering cost is $50 and holding cost is 20% of the value of the item. Find the EOQ, the

optimal number of orders, and the total inventory costs.

Given

D: annual dollar value of demand = $200,000

K: ordering cost, in dollars = $50

H: Holding cost = 0.2 (i.e. 20%)

Solution

Using EOQ formulas

Thus the optimal policy is to order $10,000 worth of supplies at a time.

Because the yearly demand is $200,000 there will be 20 orders per year

The total annual inventory costs are

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Quantity Discounts

Sellers frequently offer buyers a price discount for purchasing large quantities. There

may be several price intervals or price breaks such as $10 each unit for quantities up to

99, $9 each unit for 100 to 499, $8 each unit for 500 up to 999, and $7 each unit for 1000

and over.

The practice of quantity discounts is widely spread since it offers advantages to both the

buyer and the seller. These are listed below together with some possible disadvantages.

Advantages Disadvantages

Buyer Buyer

Lower unit price

Less paper work

Cheaper transportation

Fewer stock outs

Security (against factors like strikes, prices increases)

Uniform goods (coming from same shipment)

Larger inventories

Higher holding costs

Risk of deteriorations and obsolescence

Older stock on hand

Seller Seller

Cheaper transportation

Less paper work

Large production runs thus lower production cost per

unit

Lower unit changes

Less bargaining power with buyers

We distinguish 2 cases of discounting

a) A discount is offered at one price level

b) A discount is offered at several levels (price break)

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Example 1: Discount offered at one level

A city uses 100 replacement lamps a month for its street lights. Each lamp costs the city

$8. Ordering costs are estimated at $27 per order and the holding costs (primarily the

cost of capital) are 25%.

The city currently orders according to the EOQ. The supplier has now offered the city a

2% discount if the city will buy 600 lamps at a time. Should the city accept the offer?

Solution

Given

D = 100 units per month x 12 months = 1200 units per year

per lamp per year

K = $27 per order

lamps

The current total annual inventory cost is

= 180 + 180 = $360 per year.

To this cost, we add the item cost which is relevant when discounts on the item are

considered. If we let be the item costs at the ith price break, then

Annual cost of items = CD

= $ lamps

=

Thus the total system cost is

$360 + $9600 = $9,960

Review of the discount offer

The offer to buy 600 units at a 2% discount will reduce the item cost, the holding cost

will be higher since the city will buy 600 units. The analysis is shown in the table

below:-

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No discount

= 180

K = 27

D = 1200

H = 2.00

Discount

Q = 600 (given)

K = 27

D = 1200

= 1.96 (2% less than previous H)

Total annual ordering cost = 180

Total annual holding cost = 180

Total annual unit cost = 9600

Total annual ordering cost =

Total annual holding cost =

Total annual unit cost (2% off) = 9406

Total cost = 9960 Total cost = 10,050

Conclusion: The discount offer should be rejected. The city will be at a disadvantage to

accept it. A higher discount rate should be negotiated instead (e.g. a 5% discount is

favorable).

Example 2: Discounts at several price breaks

A general hospital buys a certain antibiotic from a certain supplier. The drug can be

bought at the following prices:

For quantities of 1 up to 4,999 - $ 2.75 a unit

For quantities from 5000 to 9,999 - $ 2.60 a unit

For quantities over 10,000 units - $2.50 a unit

The demand (D) for the drug in the hospital is 50,000 units per year. There is an ordering

cost (K) of $50 per order and a holding cost of 20% of the cost of the item per unit per

year. The problem is to find the optimal purchasing policy for the hospital.

Solution

We find the EOQ (labeled ) for the lowest price level ($2.50 in our case)

Units

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(Note that H = 20% of 2.50 = 0.2(2.50) = $0.5)

The price $2.50 is for the range of over 10,000. Since is not within this range, the

solution is not feasible.

Select the next higher item cost ($2.60 in this example) and calculate

Units

The price of $2.60 is for the range of 5000 – 9,999. Since is not within the range, the

solution is not feasible.

Compute the EOQ for the next higher price ($ 2.75 in the example)

Units

is within the appropriate range for the price of $2.75. Therefore it is a feasible

solution.

Then we compute the total annual cost for the feasible EOQ and compare it with the total

annual cost of each of the minimum quantities required for each price break. The total

annual cost is computed as

TC = ordering cost + holding cost + unit cost

TC1 (for 10,000)

TC2 (for 5,000)

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TC3 (for 3,015)

Therefore an order of 10,000 units at a time should be placed since it exhibits the lowest

total cost of $127,750 per year.

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