10.
INVENTORY MODELS
10.1. INTRODUCTION
In order to prevent the negative effects of monopolization and artificial price increases
in the market, governments go to stockpile goods that are needed by the people. On the other hand, all
organizations consider it necessary to maintain inventory of goods for their own needs or to provide
to customers outside the organization.
Policies about how much inventory to keep on hand are not new. With the quantitative
techniques developed by theorists and practitioners, the inventory policy became more scientific.
The main purpose of the inventory policy is to provide the most useful service to those who use the
inventory or, in a sense, to realize the lowest cost for the desired level of service. As it can be
understood from this expression, keeping inventory on hand provides benefits as well as increases the
cost. The inventory models to be discussed at work help determine what amount of inventory should be
ordered and when orders should be made.
Inventory is defined as goods held for future production and sales. There are
advantages and disadvantages to having too much or too little inventory on hand. Once the on-hand
inventory is low, the firm or business can always face out of stock. The damage caused by running out
of stocks is seen as consumer dissatisfaction or loss in sales. In addition, if the stock is low, the late
arrival of the raw material orders to the enterprise causes the production to pause. Keeping inventories
would also incur a certain cost, and cash invested in stocks could yield profits if invested
elsewhere. Here, the most appropriate solution of all these problems will be tried to be provided with
the inventory models that will be discussed.
10.2. INVENTORY COSTS AND RELATED VARIABLES
For businesses , the monetary aspect of inventories is important in determining the
inventory policy. Because of the inventory, the business has to endure various financial
difficulties. The purpose of inventory models is to keep the stock level that will provide the
lowest cost.
The costs included in the models to be discussed in the following sections are:
a) Purchase Cost: The price actually paid to the source from which the ordered
good is purchased . This price is the purchase price and is indicated by (k).
b) Order Cost: Costs caused by ordered inventories and can be divided according
to whether they are dependent on the order quantity or not. In the inventory
models to be considered , there will be no distinction between order costs .
Ordering costs that depend on the order quantity are the costs that depend on the
order quantity. These costs decrease as the order quantity increases and increase
as it decreases. Ordering costs include fixed transportation costs, pick-up costs,
labor charges , inspection and registration costs, costs of unpackaged goods. It
includes costs such as stamps, paper, printing, postal and telephone .1 In our
topic, order costs are shown with the v symbol .
c) Cost of Carrying: As we mentioned in the introduction, a cost arises with
keeping stocks . This cost usually depends on many factors. We will consider
a few of them . The first factor can be called the cost of capital factor . Cash
invested in inventories usually incurs an interest cost when borrowed . On the
other hand , since the cash on hand is invested in stocks, it cannot be used
for income-generating investments in other fields . Thus, an opportunity cost is
encountered. Opportunity cost means giving up a certain amount of other goods
and services in order to produce any good or service .2
Other factors that determine the cost of carrying can be listed as follows :
1. Losses due to physical deterioration, wear and being out of fashion in case of
going to stock,
2. Costs related to maintenance, processing, registration and counting of inventories,
3. Insurance costs: Costs incurred to insure inventories against the risks they
will face,
4. Taxes: Taxes that local governments or government impose on stocks .
5. Storage cost: Rents paid to store inventories.
The symbol for inventory carrying cost is c.
d) Inventory Depletion Cost: When a demand occurs, if the available goods are not
sufficient, there will be a cost in terms of business if there is no stock of goods to
meet the new demand. This cost is called stock-out or out-of-stock cost.
Inventory depletion affects producers in two ways. The first is the postponement
of consumer demands, that is, of sales . This situation creates a cost
(communication costs to inform the customer, wages paid to people working in
the warehouse, compensations, etc.). Secondly, the consumers meet their
demands from another place, which causes loss of sales to the business.' The cost of
depletion of stock will be denoted by (r) in our work .
Inventory problems can also be classified according to the intermittent
presentation, ordering, and ordering decisions.
The demand will be treated as a quantity denoted by D in the demand time division.
For example, the demand amount for a product may be D=2400 units/year. In
this case, a time period of 1 year was chosen. 1 month (1/12 years) can also be selected as
the time period and the demand amount can also be shown as D=200 units/month.
No matter what time unit is used to measure the demand the actual claim
amount should remain the same. Usually the amount of demand is known, or the
known probability distribution works. However, in some cases, the amount of demand may
not be known, as is the case for new products.
Lead Time: There is usually a period of time between when an order is
placed and when goods are received. This time period is called lead time and will be
denoted by the variable L. Therefore, if L=10 days and we want to receive the goods on the
25th of May, the orders must be placed on the 15th of May. If both the demand quantity
and the lead time are known, the demand in the lead time period is known with
certainty and can be decided without any hindrance. The lead time can be fixed and
variable.
Production Speed: If the goods we want to buy are ordered from the presenter
or the vendor, the desired goods are delivered at once by truck, train, mail, etc. reaches us.
On the other hand if the goods are produced in our business. A situation may arise
where goods are “in series” off the production line and properly entered into the
inventory count during the flow of the production process.
In inventory models, the inventory entry speed of the goods is an important factor
and will be shown with P as the unit in terms of time.
For example, if 200 units are produced per day while the factory is in
production, it would be equivalent to 200 units/day x 30 days/month= 6000 units/
month or 6000x12= 72.000 units/year per month. Therefore, P is a velocity like the
velocity of the claim.
Other Controllable Variables:
Two variables will be mentioned here.
1. Number of orders over time: Although the inventory policy reports the order
quantity and the reorder point, it is also useful to know how many times an order is
made per year. The order number will be indicated by (n).
For example, if D=2400 units/year and we order 200 units each time, the
number of orders per year would be (2400 units / year / 200 units) = 12 orders / year
or 1 order / month.
2. Time between orders: The time between orders (t) specifies how often the order is
made, like the variable n. In fact, there is a very close relationship between t and n. If we
place 18 orders in a year, this means that 2 orders in 10 days or orders are made once in 1/12
years, which is the same thing.
In this section, deterministic and stochastic inventory models are discussed in various
situations and how they can help in inventory control policy is tried to be explained with
examples.
10.3. DETERMINISTIC INVENTORY MODELS
In this section, models whose parameters are known and which also include
stock depletion under full certainty will be discussed. The main purpose behind
creating the inventory model is to determine the optimal (ie, the lowest cost) values
for the decision variables (how much, when). Accordingly, optimization aims to find the
value of the decision variables that will minimize the annual total cost. So the first model to
be considered will be the 'Basic Economic Order Quantity' model.
10.3.1. Basic Economic Order Quantity Model
This model includes the following assumptions.
1. The demand per period is definite and the rate of demand is fixed.
2. Goods are ordered at equal intervals.
3. The price of goods is fixed.
4. The transportation of the ordered goods is instant.
5. There is no out of stock situation.
6. The lead time is precisely known and is zero.
The situation that emerges after these assumptions is as follows. Orders are placed
at once and when stocks reach zero level. The inventory model that explains this situation is
shown in Figure 10.1.
The total cost of the model is the sum of the three cost combinations.
Annual Total Cost = Annual Ordering Cost + Annual Carrying Cost + Annual Cost of Goods Purchased
Figure 10.1.
Basic Economic Order Quantity Inventory Model by Time
The Annual Total cost curve is drawn by considering the three cost combinations
Figure 10.2. The inventory costs versus order quantity with emphasis on
the total cost, ordering cost, and the carrying cost
As can be seen in Figure 10.2, the optimum economic order quantity and the lowest
cost are determined.
Parameters of the model: In order to find the total cost per year or per period, the three cost components we
mentioned earlier must be determined on a period or yearly basis.
k = Unit price of the purchased good
v = Order cost regardless of order quantity Orders per period = n = D/Q
c = Cost of carrying Order cost per period = v.D/Q
D= Demand amount Average inventory level per period = Q/2
Q= Order quantity Carrying cost per period = c.Q/2
The values of k, v, c, and D are known and fixed. The carrying cost is determined according to the average inventory level per period.
Cost of goods purchased per period = k.D
Q will be the economic order quantity to be determined .
Total Cost (TC) = vD/Q + cQ/2 + kD
Since the economic order quantity Q is to be determined, in order to find Q, the
first derivative of the TC function with respect to Q must be taken and set to zero. In
addition, if the second derivative of TC is positive, the amount of Q found is the value that
minimizes the total cost.
If the economic order quantity Q is substituted into the total cost per period equation, we
can find the total cost per period.
(2)
EXAMPLE 10.1
Clothing and ready-to-wear store requests 4000 meters of fabric from Yildiz fabric
factory per month. The fabric is $300 per meter and the pay per order is $1200. The
carrying cost is 25% of the purchase price of the fabric. The clothing store operates 24
days a month. According to this: Find
a) The economic order quantity of the
store,
b) the time between orders,
c) Annual total cost.
SOLUTION
a) The time period used is one month, the parameters of the problem are;
k=$300 v=$1200 c=300(0.25)=$75 D=4000 meters/month.
Unit Cost Ordering Cost Carrying/Holding Demand rate
Cost
By using the formula (1), the economic order amount of the store we find.
Q = 357.8 meters
b) The time between orders (t) = Q/D = 0.09 months
t = 0.09 x 24 = 2.16 days
c) The annual total cost of the store is found by the formula (2).
EXAMPLE 10.2
The television seller requests 85 televisions per month from the manufacturer. The cost of each order of
the television seller from the manufacturer is fixed, $ 40,000 , and the price paid for each television is $ 120,000.
Ordered televisions come 5 times a day and stock is not allowed to run out. The carrying cost averages $450 per
stock unit per month.
It is assumed that there are 30 days in a month. According to this, find:
a) Economic order quantity.
b) The monthly total cost,
c) The time between orders.
d) The highest level of inventory
SOLUTION
Parameters of the problem
D=85 units/month, c= $450, v= $40.000, k= $120.000 Since 5 units of television are delivered per
day, the transportation speed or production rate per month becomes P=5x30=150 units.
EXAMPLE 10.3
The raw material demand for the production of an enterprise is 16,000 tons per year and
the production rate of the enterprise is 2400 tons per month. The cost of preparing or ordering a
ton of goods is $400 and the carrying cost of one tonne of raw materials per month is $10. The
purchase price of one ton of goods is $200.
Accordingly, find the
a)Optimum production amount,
b) Annual total cost,
c) Total time between preparations,
d) the highest inventory level.
SOLUTION
SO
a) The annual optimum production amount of the enterprise is calculated with the 3rd formula. One year equals 12 months
year
