Abstract – This paper helps to investigate the application of
inventory control model in Khyber Marble dealers. An Economic Order Quantity
model was used to determine the optimal order quantity in an optimal time for
multiple products. The result of the analysis shows the variations in the
stocks. The orders at different time interval was carefully and regularly updated
and verified because reorder points and EOQ may change. It was noted that
reorder points may often coincides little close together. This EOQ model can help in purchasing decisions in
real world situations more dynamically.

 

            Key Words – Economic Order Quantity, Marble Dealers, optimal
time, inventory, Re-order point.

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I.           
Introduction

            Inventory is a quantitative control
technique meaning a store of goods or a stock consisting of dependent and
independent demands. The function of inventory is to meet foreseen demands, to
smooth production requirements, to decouple operations and to protect against
stock outs. It also take advantage of expected
number of orders. It helps to hedge against price increases. It permit
operations and take advantage of quantity discounts 1.

An inventory problem occurs when there is need to stock
goods or commodities in order to satisfy demand over a specified period of
time. Almost every business must stock goods to ensure smooth and efficient
handlings of its operations. The demand to stock goods may be satisfied by
stocking once for the period of time concerned or by stocking gradually for
every time during the time horizon 2.

The EOQ
model is a basic model which is used to manage inventory levels and ordering. EOQ
model can be used in many different scenarios, but there are assumptions and
conditions that also have to be met. An EOQ model is acceptable when the actual EOQ
value is greater than the standard deviation of annual demand 3.

‘Khyber Marble
Dealers’ is selected to get details about inventory items and other products.
It is a private whole sale dealer which is run by Jahan Khan. It is located on
service road, near Sanghar town chaklala, Rawalpindi. Khyber Wholesale dealers take
the demand from customers, places their orders to the manufacturer Meherban
Khan and provide customers their finished product. Finished products are
shipped to the customer by shipper Hameed Khan. The product categories which
are presented are sunny grey (1″, ½”), sunny white (1″, ½”) and black (1″, ½”)
marble.

 

Table
I?I

DETAILS
ABOUT MANUFACTURER AND SHIPPER

Manufacturer

Name

Address

Meherban Khan

Near small industries Road,
Mardan

Shipper

Hameed Khan

Service road,
near Sanghar town Chaklala,   Rawalpindi

               

 

Table
I?II

PRODUCT
CATEGORIES

Product Name

Category Name

Sunny Grey

1″

Sunny Grey

½”

Sunny White

1″

Sunny White

½”

Black

1″

Black

½”

The study is aimed to achieve the
following objectives; to handle details about
inventory and the products, interface to generate sales order for customers and
track down the customer orders. It generate inventory control models.

 

                                                                                                                         
II.           
Literature Review

            Inventory model is defined as mathematical equation
or formula that helps an organization in determining the economic order
quantity, and the rate of ordering, to keep stocks, or services which are flowing
to the customers without interruption or delay. Inventory planning and control
is an inclusive exercise which comprises of the policies, procedures, programs
and strategies which an organization uses to regulate the range of items held
in stock, and the amount of replenishment stock required for efficient
operation. An
inventory system monitors inventory levels through inventory policies and controls
identified by the organization. Through the inventory system the
levels of inventory that has to be maintained, when inventory replenishments
should occur and the size of the orders is established. Inventory theory is used by manager
that are established rules that can be used to maintain inventory levels at the
lowest cost possible while meeting customer demand. In an inventory system
there are three types of costs that are subject to control, at least two has to
be controlled at a particular time; 1) Replenishment cost of inventory
( also called setup or order cost), 2) Carrying cost of inventory (also known
as holding cost) 3) Shortage cost 3.

Inventories
cannot be decoupled or separated from other functions such as purchasing
production and marketing. It keeps the stock levels down to make cash available
for other purposes. Inventory control also balances the conflicting goals.
Purchasing manager may wish to order large batches to get volume discounts. The
production manager similarly wants long production runs to avoid time consuming
setups. He also prefers to have large raw material inventory to avoid stops in
production due to missing materials. The marketing manager would like to have a
high stock of finished goods to be able to provide customers high service level
4.

Another
goal of inventory control model is to determine the optimal quantity of objective
to order for and then to place an order given the certain contingencies and
peculiarities of a particular business scenario.

Further
studies shows that an inventory models has to be developed for the smooth
running process in determining stock control in an organization. In
probabilistic inventory models, Single-period
inventory model is one of those elementary models in which only a single
procurement is being made 5. Application of such model can be seen in production management
systems, for example stocking seasonal items.

An EOQ model is adequate
when the standard deviation of the annual demand is smaller than the value of actual
EOQ. Literature
shows that standard economic order quantity (EOQ) model is based on the
assumption that the dealer’s capitals are unrestricted and must pay for the products
as soon as the products are received. However, this may not be true in every
case. In most of the real cases, the supplier allows up to a certain fixed
credit period to settle the account for stimulating dealer’s demand. During the
credit period before payment must be made, the retailer can sell the items,
accumulate sales revenue and earn interest. Goyal was the first one who developed
an economic order quantity (EOQ) model under the conditions of accepted delay
in payments 6.

In Fixed-Order Economic Order Quantity Model the order
quantity stays constant. The inventory levels are monitored and when the stock
level reaches a specific point an order is placed. The only danger regarding
stock depleting to zero is during lead time. The demand range can be determined
by the analysis of past demand data. Safety stock is dependent of the service
level desired, the safety stock must not be too high as this will incur excess
inventory cost. In the multiproduct EOQ system products of
differ frequency usage are set alongside each other and comparisons regarding
each item’s reorder frequency is made. Reordering cycles of the slower items
are lined up to that of the fast moving items and corresponding order cycles
can be determined.

Combination
of a fixed-order EOQ model and multi-product EOQ
model was most probably considered sufficient. It was also found that reorder
points and EOQ may change. In EOQ models, it was observed that the reorder
points sometimes coincide or occur close together. An EOQ model is acceptable
when the standard deviation of the annual demand is smaller than the actual EOQ
value 3.

Literature
shows multi-product production and inventory model using the net present value.
It was also noted that as long as deterministic demand is used, the two models shows
very little variations, but, while dealing with the probabilistic or stochastic
demand, the differences are major. The EOQ model uses long-run average cost and
is fairly simple as compared to the complexity of using net present value. the
results are also straight forward and robust 2.

Base-stock
policy was used by some researchers in addressing the problem of determining
the optimal production and inventory policy for a multi-product production
inventory system. The problem which arose concerns homogenous products with an
average determined demand for each time period and random demand during the
period over an infinite horizon. To investigate the effect of optimal
production and inventory policy when the products are not homogenous, a
heuristic algorithm had to be generated. This developed into a very complex
algorithm that had to be implemented on each product 7.

EOQ
model is not always considered as adequate for slow and fast moving items in a
system. In general, it can be understood that an EOQ model is fully acceptable
for medium and slow moving items in a system.

 

                                                                                                                                               
III.           
  Method

The basic EOQ model is used for
determining the optimal order size that minimizes the sum of carrying costs and
ordering costs. The model formula is derived under a set of restrictive
assumptions, as follows:

Demand is known with
certainty and is constant over time.
No shortages are
allowed.
Lead time for the
receipt of orders is constant.
The order quantity is
received all at once.

The two costs i.e. carrying and ordering costs can react
inversely to each other. As the order size increases, less orders are required,
resulting the ordering cost to decline, whereas the average amount of inventory
on hand will increase, resulting in an increase in carrying costs. Thus, the
optimal order quantity represents a cooperation between these two inversely
related holding and setup costs.

Example: The
I-75 Carpet Discount Store in North Georgia stocks carpet in its warehouse and
sells it through an adjoining showroom. The store keeps several brands and
styles of carpet in stock; however, its biggest seller is Super Shag carpet.
The store wants to determine the optimal order size and total inventory cost for this
brand of carpet given an estimated annual demand of 10,000 yards of carpet, an
annual carrying cost of $0.75 per yard, and an ordering cost of $150. The store
would also like to know the number of orders that will be made annually and the
time between orders (i.e., the order cycle) given that the store is open every
day except Sunday, Thanksgiving Day, and Christmas Day (which is not on a
Sunday).

Solution:

Cc = $0.75 per yard

Co = $150

D = 10,000 yards

The optimal order size is

The total annual inventory cost is determined by substituting Qopt into
the total cost formula:

The number of orders per year
is computed as follows:

Given that the store is open
311 days annually (365 days minus 52 Sundays, Thanksgiving, and Christmas), the
order cycle is

This example clearly explains one of the inventory
control model which is economic order quantity model which calculates the
optimal size in an optimal time. The specific formulas for the application of
economic order quantity EOQ (Q*), total costs (TC), expected number of orders
(N) and expected time between orders (T) were also demonstrated by an example
mentioned above. Further, if reorder point is to be calculated, demand per day
will be multiplied by the lead time in order to calculate product’s ROP. Demand
per day can be calculated when annual demand is divided by the total number of
working days in a year.

 

In ‘Khyber
Marble Dealers’ three products that is sunny grey, sunny white and black having
two categories (1″ and ½”) respectively are selected for the inventory control
model analysis.

Real time data of
seven customers is collected who placed their orders for marble to be delivered
to their location on time which is shown in Table
III?II. This
real time data is collected to generate the specific optimal model, having no
initial inventory, in order to calculate the optimal order quantity at the
optimal time, taking into consideration the inventory level of items and lead
time by optimizing the frequency of shipment of orders. Table
III?I shows
the details of marble Dealers.

 

Table III?I

MARBLE
DETAILS

Product Name and category
(Sq. ft.)

Sale price per unit

Purchase price per unit

Setup cost (sq. per ft.)

Holding cost (sq. per ft.)

Sunny Grey ½’

28

24

400

0.2

Sunny Grey 1′

50

43

400

0.3

Sunny White ½’

34

30

400

0.2

Sunny White 1′

62

56

400

0.3

Black ½’

190

160

400

0.2

Black 1′

320

280

400

0.3

 

Table III?II

CUSTOMERS
DETAILS

Sr. No.

Customers

Address

1

Anayat Ullah

Hs#68,
St#7, Mangral Model Town, Rawalpindi

2

Anwar Ul Haq kiani

Hs#26,
St#1-A, New Mangral Model Town, Rawalpindi

3

Raja Abid

St#18-B,
Sultan Town, Rawalpindi

4

Raja Sami

St#23,
Sabri Street Chaklala, Rawalpindi

5

Zahid Bhatti

Hs#56,
St#13, Shaheen Town, Rawapindi

6

Raja Ghulfam

Near
Leading Public School, Shaheen Town, Rawalpindi

7

Usman Aslam

Jamia
Farooq Masjid, Rawalpindi

 

Table III?III and Table
III?IV shows
the calculations for inventory control EOQ (Q*) model, total costs, expected
numbers of orders and expected time between orders, holding and ordering costs
when no discount is provided to the customers (that is, sales price per unit). Given
that the store is open 308 days annually (365 days minus 52 Sundays and Islamic
festival Holidays).

 

Table III?III

CALCULATIONS
FOR EOQ MODEL

Demand
(Sq. ft.)

Annual Holding cost

EOQ (Q*)

Expected No of Orders (N)

Expected time between
orders (T) (days)

13400

5.6

978

14

22

2650

15

266

10

31

9600

6.5

751

13

24

2200

18.6

218

10

30

1540

38

127

12

25

190

96

28

7

46

 

Table III?IV

CALCULATIONS
FOR EOQ MODEL

Ordering cost

Carrying cost

Total cost (Rp.)

5478.69

2739.34

8218.03

3987.48

1993.74

5981.22

5109.99

2555.00

7664.99

4045.74

2022.87

6068.61

4838.18

2419.09

7257.27

2701.11

1350.56

4051.67

The
EOQ will sometimes change when there is a quantity discounts offered by some
suppliers as an incentive to customers who places larger orders. Table
III?V shows
the calculations of model when discount is given to the customer. When holding
cost is expressed to be 10% for ½’ marble and 15% for 1′ marble, Total cost is then
reduced when this discount is given to the customers.

Table III?V

CALCULATIONS FOR EOQ MODEL WHEN DISCOUNT IS GIVEN

Product Name and category
(Sq. ft.)

Annual holding cost

EOQ (Q*)

No of Orders (N)

T days

Total Cost (Rp.)

Sunny Grey ½’

2.8

1384

10

32

5811.02

Sunny Grey 1′

7.5

376

7

44

4229.36

Sunny White ½’

3.4

1063

9

34

5419.96

Sunny White 1′

9.3

308

7

43

4291.15

Black ½’

19

180

9

36

5131.67

Black 1′

48

40

5

65

2864.96

Table III?VI shows the calculations for Reorder point. Demand per
day (d) is the slope which is the ratio of demand to number of working days in
a year.

 

Table III?VI

CALCULATIONS
FOR REORDER POINT (ROP)

Product Name and category
(Sq. ft.)

d (demand per day)

Lead Time (days)

ROP

Sunny Grey ½”

43.5

3

130.5

Sunny Grey 1″

8.6

3

25.8

Sunny White ½”

31.2

3

93.5

Sunny White 1″

7.1

4

28.6

Black ½”

5.0

5

25.0

Black 1″

0.6

5

3.1

 

                                                                                                                     
IV.           
Results/Discussions

It was noticed
that Khyber marble dealers does not have a proper inventory management which
leads them to sometimes under stocking. The decision was made to develop a
Multi-product Economic Order Quantity model to determine the optimal order
quantities, numbers of cycles and order times which were calculated to be as
follows (Table IV?I). The rate at
which different products like sunny grey, sunny white and black marble can be
ordered were determined.

When holding
cost is reduced to 15% and 10% for 1′ and ½’ respectively, total cost is
reduced and gives a maximum change in an EOQ model, expected time and number of
orders. This change is illustrated in Table IV?II.

 

Table IV?I

OPTIMAL MODEL

EOQ (Q*)

Expected No of Orders (N)

Expected time between
orders (T) (days)

Total cost (Rp.)

978

14

22

8218.03

266

10

31

5981.22

751

13

24

7664.99

218

10

30

6068.61

127

12

25

7257.27

28

7

46

4051.67

 

Table IV?II

OPTIMAL
MODEL WITH GIVEN DISCOUNT

EOQ
(Q*)

No of
Orders (N)

T (days)

Total
Cost (Rp.)

1384

10

32

5811.02

376

7

44

4229.36

1063

9

34

5419.96

308

7

43

4291.15

180

9

36

5131.67

40

5

65

2864.96

 

                                                                                                                                            
V.           
Conclusion

Inventory or stock control is a quantitative control
technique with strong financial implication. An Economic Order Quantity model for
multiple products were used to determine the optimal order times. The orders at
different time interval were constantly updated and verified because reorder
points and EOQ may change. To determine whether it makes sense to take the advantage of a quantity
discount an owner of any business must compute the EOQ and compute the total
cost of inventory for the EOQ and then select the order quantity that delivers
the minimum total cost.

Further, Inventory management was the main issue which Khyber marble dealers
were facing, for that reason Economic order quantity was applied in order to
meet the timely delivery of the products to the customers. By the implication
of this model, Understocking of products were reduced. The flow of process
becomes good. Total costs were reduced resulting an increase in Profit. EOQ
concept when applied to the real world situations help in purchasing decisions
more dynamically.