Inventory models

Engineering Management 19/12/2013

M M HASAN, Lecturer, AIE, HSTU, DINAJPUR 1

MD MOUDUD HASAN LECTURER AIE, HSTU DINAJPUR

An inventory problem exists when it is necessary to stock physical goods or commodities for the purpose of satisfying demand over a specified time horizon(finite or infinite).

Almost every business must stock goods to ensure smooth and efficient running of its operation.

Decisions regarding how much and when to order are typical of every inventory problem.



1. Economic factors: a) Setup cost:

• Fixed charge associated with the placement of an order

• Independent of the quantity ordered

b) Purchase price: • Parameter is of special interest when quantity discounts

or price breaks can be secured

c) Selling price: • Unit selling price may be constant or variable depending

on whether quantity discount is allowed.

d) Holding cost: • The cost of carrying inventory in storage

• It includes the interest on invested capital, storage, handling cost, depreciation cost, etc.

e. Shortage cost: • The penalty cost incurred as a result of running out of

stock when the commodity is needed.

• Include costs due to loss in customers goodwill and potential loss in income.

2. Demand: a) Deterministic demand : • It is assumed that the quantities needed over

subsequent periods of time are known with certainty.

b) Probabilistic demand: • The requirements over a certain period of time are not

known with certainty

• Requirements pattern can be described by a known probability distribution



3. Ordering Cycle: a) Continuous review • A record of the inventory level is updated continuously

until a certain lower limit is reached at which point a new order is placed.

b) Periodic review • Orders are placed usually at equally spaced intervals of

time.

4. Delivery Lags or Lead Times: • Time between the placement of an order and its

receipt is called delivery lag or lead time.

5. Stock Replenishment: • The actual replenishment of stock may occur

instantaneously or uniformly.

6. Time Horizon: • The time horizon defines the period over which the

inventory level will be controlled.

7. Number of supply Echelons: • An inventory system may consist of several stocking

points.

8. Number of items: • An inventory system may involve more than one

item.



The simplest type of inventory model occurs when demand is constant over time with instantaneous replenishment and no shortages.

Typical situations to which this model may apply are- 1. Use of light bulbs in a building.

2. Use of clerical supplies, such as paper, pads, and pencils, in an office.

3. Use of certain industrial supplies such as bolts and nuts.

etc.



Let, K= Setup cost per unit time

h= Holding cost per unit inventory per unit time

β= Demand per unit time

Y= Inventory level

TCU= Total cost per unit time

to=Y/β

TCU as a function of Y can be written as

TCU (Y) = (Setup cost/ unit time)+(holding cost/unit time)

= ��

�

+ ℎ ×�

�

TCU(Y)=��

�+

��

�

For minimum cost, �

�� = 0

so that, −��

��+

�

�= 0

⟹�

�=

��

��

⟹ �∗ =��

�

This is called as wilson’s economic lot size for single item static model.



Total inventory cost,

�� ∗ = 2��ℎ

• Assumptions in single item static model: I. Single item

II. Deterministic demand

III. Instantaneous replenishment

IV. Zero delivery lag

V. Uniform demand per unit time.

The purchasing price per unit depends on the quantity purchased.

This usually occurs in the form of discrete price breaks or quantity discounts.

In such cases, the purchasing price should be considered in the inventory model.

Consider the inventory model with instantaneous stock replenishment and no shortage.

Let, C1= cost per unit for y<q

C2= cost per unit for y≥q

The total cost per unit time for y<q is

�� = ��1 +��

�+

ℎ

2�



• The total cost per unit time for y≥q is

�� = ��2 +��

�+

ℎ

2�

• Let Ym be the quantity at which the minimum values of TCU1 and TCU2 occur.

�� =2��

ℎ

Define y=q1(>ym) such that

TCU1 (ym) = TCU2(q1)



Total cost per unit

�� ⋇ =

�� 2 �� , � ≤ ��

�� , �� < � < �� 1

�� , � ≥ �� > ��

�� ⋇ = �� , � ≤ �� ⋇ = �� , �� < � < ��

�� ⋇ = �� , � ≥ �� > ��

q1 q1

q1



The daily demand for a commodity is approximately 100 unit. Every time an order is placed, a fixed cost Tk. 100 is incurred. The daily holding cost per unit inventory is TK. 0.02. if the lead time is 12 days . Determine the willson’s economic lot size and the reorder point.

Solution:

Given,

Demand, β= 100 unit/day

Setup cost, K= 100 Tk/order

Holding cost, h= 0.02 Tk/unit/day

Lead time= 12 days

We know,

Economic lot size, �∗ =��

�=

�×��×��

�.��= 1000 ��

Required, i. Economic lot size =? ii. Reorder point=?

Ordering time,

�∗ =�∗

�=1000

100= 10 ��

Effective Lead time= 12-10=2 days

Demand for effective lead time= 2β = 2 x 100 =200 unit

Reorder point= 200 unit

10 days

1000 unit

Reorder point

Lead time 12

days

200 unit



Consider the inventory model with the following information.

Demand, β= 5 unit/day

Setup cost, K= 10 Tk/order

Holding cost, h= 1 Tk/unit/day

C1=2, 0<q<15

C2= 1, q≥15

We know,

First compute, �� =��

�=

�×��×�

�= 10 ��

Since ym< q, it is necessary to check whether q is less than q1.

The value of q1 is computed from

TCU1 (ym) = TCU2(q1)

��1 +��

��+

ℎ

2�� = ��2 +

��

��+

ℎ

2��

⟹ 5× 2+10× 5

10+

1

2× 10 = 5× 1+

10× 5

��+

1

2��

⟹ ��

2 − 30�� + 100 = 0

⟹ �� =30± 900− 400

2

⟹ �� = 26.18 �� 3.82

By definition, q1 is selected as the larger value.

Since q1 > q, it follows that y*=q=15

Total cost per unit,�� ⋇ = �� , �� < � < ��

=� × � +��×�

��+

�

�× ��

= 15.83

Education

Inventory models