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#### Formulation of Optimal Economic Order Quantity under Different Inventory Models

Dr. Chittaranjan Mallick

Department of Mathematics, Parala Maharaja Engineering College, Sitalapali, Berhampur, 761003, B.P.U.T., Odisha, India

ABSTRACT

Decision making in production planning and inventory management is basically a problem of coping with large numbers and with a diversity of factors external and internal to the organization. The formulation and solution of an inventory model depend on the demand per unit time of an item which may be viewed as deterministic or probabilistic. The classical economic order quantity (EOQ) model assumes that items produced are of perfect, quality and that the unit cost production is independent of demand. However in realistic situation, product quality is never perfect, but it is directly affected by the reliability of the production process. Invertories are idel resources that possess economic value. There are vitally important to manufacturing firms because they store the value of the labour and processing activities used to make their products. Adequate inventories facilitate production and help to assure customers of good service, on the other hand carrying inventories ties up working capital on goods that sit ideal, and earning any return on investment. Hence the role of inventory management is to maintain adequate, but not excessive levels of inventories. The objective of basic inventory model is to determine the optimal order quantity that minimise the total incremental costs of holding

inventory and processing orders. In general the main objective of the inventory model is to minimize the cost of production and minimize the profit. In the present paper we have made an attempt of derive an economic order quantity formula in the different situations prevailing on the average cost of the item along with demand rate keeping finite production associated with the stock position. While analysing the problem we have formulated the EOQ for an optimal order level and the minimum average cost with uniform demand satisfying the shortage criterion.

Key word : Inventory model, EOQ model, Setup Cost, Shortage Cost.

Introduction :

Inventory plays very important roles in decision science, economic, optimization design Science, Communication, Supply chain management, news vender problem, managements in style goods and perishable items and many other fields of science and technology. It can be considered as a necessary evil because the lack of synchronization in the production system makes holding inventory indispensable. Generally inventories deals with various functions such as : (i) Co-ordinating operations (ii) Smoothing production (iii)

Achieving economics of scale and (iv) Improving customer service. The problem which helps to determine the most optimal order quantity under stable conditions is generally known as classical economic order quantity (EOQ) inventory problem.

Though the theory of inventory have been studied more than a century, yet this field of study is drawing more attentions from the researchers of many other branches of mathematics. The main reason behind this attraction is to search a suitable replacement for the decision making for out requirement, which is essential to solve many real world optimization problems.

However on the cases of discussion it may be described briefly as An inventory can be classified into raw materials inventory, work- in process inventory and finished goods inventroy. The raw materials inventory removes dependency between supplliers and plants. The work in process inventory removes dependency between various machines of a product line. The finished good inventory removes dependency between plants and its customers or market. The inventory model is primarily concerned with answering two fundamental questions :

How much to order ?

When to order ?

Total P u rchasin g Setup H old in g Shortage

Over the years, a voluminous amount of

research on this subject has been done and many

in ven tory = co st

co st

+ cost +

cost

+ c os t

interesting results have been published in various journals. Extensive survey of research work have been conducted by Bishop [1], Prasad [2], Zipkin [3], Spencer [4], Silver [5],

urgreletti [6], Wanger [7] and Ziegler [8]. Rosenblatt and Lee [9] have discussued two sophisticated EOQ model with imperfect production which deteriorate over time under different scenarios.

The setup cost of the items produced is an important factor in EOQ model. First of all Kaplan [10] points out that there are three fundamental purposes for cost accounting system such as valuing inventory for financial statement, providing feedback to production managers and measuring the cost of individual stack keeping unit (SKU).

It is a natural curiosity to know, what decisions are requirement ? And why one should replace it ! The objective of this presentation is not to describe these two questions vividly.

The answer to the first question determines the economic order quantity (EOQ) or Economic lot size problem or optimal order size by minimizing the following cost model.

An inventory problem exists if the amount of the goods in stock is subject to control and if there is at least one cost that decreases as inventory increases. The inventory model is based on the following facts.

Advantages :

The economics of production with large run sizes.

The smooth and efficient running of the business.

The economics in transportation.

Faster and adequate service to customers.

Profit from the market where prices are expected to rise

Disadvantages :

House rent

Interest on invested capital

Physical handling

Accounting

Depreciation and deterioration

In this paper we have generalized the work of Silver [11] and waters [12] where we have formulated economic lot size in general keeping demand rate is uniform with finite production along with shortage. Attempts also made to incorporate the economic lot size keeping different production cycle with finite production rate having no shortage.

The organisation of the article is follows : Following the introduction, section – (2) presents important notations and assumption to derive different theorems ralating EOQ. Numerical Examples and concluding remarks are presented in section 3 and 4 respectively.

Some Definitions and Notations

EOQ is that size of order which minimizes total annual (or any other time period as specified by individual firms) costs of carrying inventory and cost of ordering.

Co s ts

Total c o s t

(c) Price per unit, ordering and carrying costs are all assumed to be constant regardless of order quantity.

Purchasing Cost : Purchasing cost is based on the price per unit of the item. It may be constant or it may be offered at a discount that depends on the size of the order.

Setup Cost : Setup Cost represents the fixed charge incurred when an order is placed. This cost is independent of the size of the order.

Holding Cost : Holding cost represents the cost of maintaining the inventory in stock. It includes the interest on capital as well as the cost of storage, maintenance and handling.

Shortage cost : Shortage cost is the penalty incurred when we run out of stock. It includes potential loss of inome as well as the more subjective cost of loss in customers good will.

2.2. Notations

I = The cost of carrying one rupee in Inventory for the unit time.

hc = Holding cost per unit per unit time. Sc = Shortage cost per unit per unit time

=Setup cost per production run

M in i

total c o s t

EO Q

Inventory

ca rry in g c o st O rdering co s t

No . o f Un its

q = Lot size per production run (i.e. the quantity produced in one run)

td = Total demand dr = Demand rate

[Fig.1 – Relationship between cost & Quantity]Assumptions of EOQ.

Precise knowledge of demand of items and there uses rate are also assumed to be constant.

Delivery of units ordered is virtually instantaneous.

K = Production rate

C = Average total cost per unit time

ti = time interval between run i and i + 1 for i = 1, 2, ….n.

T = total time.

t = time interval between two consecutive ordering of inventory.

z = Stock level / order level L = Lead time

q*, t*, z* be the optional values of q, t, z

C 1

2

h q sc

c t

h q sc . dr

c q

…(2)

1

1

respectively for which the cost C is minimum. Now we will prove the following theorems.

For minimum value of C we have

dC 0

dq

Theorem – 1 :

The minimum cost per unit time of the Economic lot size is proportional to the Square

2 hc

sc dr 0

q2

root of the demand rate keeping holding and

and the minimum order is

setup cost constant.

Proof :

q*

2sc .dr

hc

…(3)

In this inventory model orders of equal size are placed at periodical intervals. The purchase price per unit is same irrespective of order size.

From (1) and (2) the optimum value of t and the minimum cost per unti time is given by

Let us suppose.

Demand is uniform at a rate of d units

t*

2sc

h .d

…(4)

r

time.

Production rate is finite.

C 1 h .

c r

2sc .dr

s .d hc

Lead time is zero

min 2 c h

c r 2s .d

2hc.sc .dc

c c r

Shortages are not allowed.

Let q be the units of quntity produced per production run at interval t.

Since the demand rate is dr units per unit time, then the total demand in one run of time interval t is dr.t

The quantity produced per production run is q = dr.t …(1)

h

Considereing hc and are constants then the minimum cost per unit time is proportional to the square root of the demand rate.

Theorem 2 :

The minimum cost per unit time of economic lot size is propertional to the square root of total demand value, the demand rate are different in different production cycles with finite production rate having no shortage.

The cost of holding inventory is

h q. t

2 C

Proof :

Let q be the units of quantity produced per

The total cost per production run of time t

is

given by C h h qt s

production run. If R is the total demand in the total period t, then the number of production

R

2 c c

cycles =

n.

q

The average total cost per production run of time t is

(since the shortages are not allowed)

Q Q Q Q Q

1 2Rs

Rs h t

C h

c c c

I n v e n to r y l e v e l

I n v e n to r y l e v e l

min 2 c

ht t 2Rs

c c

Tim e

or, C

hc sc R

…(8)

O t1 t2

As t , t , …..t

t3 t4 tn

T

Fig – 2

be the times to the successive

min 2t

It is clear that the relations (7) and (8) may be obtained from the corresponding relation of

and (4) of theorem (1) replacing dr (demand

1 2 n

production cycles than t = t1 + t2….+ tn … (5) Thus q the quantity produced at the

beginning of each production run is supplied

rate) by (average demand rate).

Theorem 3 :

The minimum cost per unit time of

with different uniform demand rates in times t1,

economic lot size is

2s

2s

q q* c

drk

t2,…..tn

in the successive cycles.

hc

k d

r

t is

The cost of holding inventoty for the period

is the economics lot size.

Proof : Let K > dr the number of items produced

h 1 qt 1 qt ….. 1 qt 1 qh t

per unit time.

c 2 1 2 2 2 n 2 c

and setup cost = ns R s

c q c

If q is the number of items produced per production run then the item produced per production is

The total cost for the period t

t q

1 k

…(9)

1 qh t R s

and the time of one complete production

= 2 c q c

…(6)

The average total cost per unit time

run i.e.,

t q dr

…(10)

1

C = qh

Rsc

If Q is the inventory level at the moment

2 c tq

the production is completed then

For minimum value of C

Q q d t q d q q

1 dr

…(11)

dC 1

Rsc

r 1 r k

k

h 0

dq 2 c

tq2

The cost holding inventory for the period

q*

2Rsc

…(7)

t h

tQ 1 qh 1 dr t

c 2 2 c k

hct

Hence (7) gives economic lot size formula.

The minimum cost per unit time is

The total cost one run of period

t s 1 qh 1 dr t

c 2 c k

The total cost per unit time Let z be the order to which the inventory is

c

c

C sc 1 qh 1 dr

raised in the beginning of a run of time interval t This inventory is reduced to zero in time

z

z

t 2 k

r

r

The total cost per unit time

t1 d

…(14)

sc .dr

1 qh

1 dr

q 2 c k

…(12)

Then the shortages arise and increase from

For minimum value of c

dc sc dr 1 h 1 dr 0

0 to Q – z in the remaining time (t – t1) where Q is the total demand for one run.

Q = dr.t …(15)

dq q2

2 c k

The cost of holding inventory

h 1 z t

2

2

hc z

q q* 2sc

drk c 1

hc

k d

r

…(13)

2 2dr

The shortage cost for one run

Using (13) the minimum cost is

= s 1 Q z .t t

C s d

hc

k d

c 2 1

min

c r 2s d k

1

1

c r

= sc

Q z .t t1

1 k d

2s kd 2

r

r

hc

c r

2 k

hc

k dr

= 1 s Q z . R z 1 sc Q z 2

2 c d d 2 d

r r r

2h .s d

1 dr

The average total cost per unit time

c c r k

and the time of one run is

C z 1 hc

z2

sc Q z 2

t 2d

r 2dr

* q* 2s k

t c

dr hcdr k dr

dr hc

z2

sc Q z 2

Theorem 4 :

Q 2dr

2dr

The minimum cost of economic lot size in constant time t with finite rate of production

c (z)

hc z2

2Q

sc Q z2

2Q

and having shortage i given by

Since the time t is fixed and known and

C hcscQ

hcscdrt

s is constant the average setup cost is also

min 2h

s

2h

s c

c c c c

Proof : To proof this theorem we assumes the condition (i), (ii), (iii) of theorem (1) and allowing shortages with backlogg

sc

constant t

C(z)

is not included on the average cost

For the minimum value of C(z)

h z2

The cost of holding inventory = c

dC hc z sc Q z 0

dr

Theshortage cost for one run

dz Q Q

s Q s d t

sc q z 2 sc d t z 2

r

r

z z* c c r

…(17)

2dr

2d r

hc sc hc sc

Which is the required optimal order level

The total cost per run

From (4) the minimum value of C since

hc

2d

z2

sc d t z 2 s

r c

r c

2d

d 2C h s r r

dz2

c c and from (3) minimum

Q

The average total cost per unit time

cost per unit time is given by

h z2 s d t z 2 s

C z, t c c r c

…(19)

h s Q s s Q 2

d s

2dr t 2dr t t

Cmin c c c Q c r c

2Q hc sc 2Q hc sc Q

Which is a function of two variables z and t for C (z, t) to be minimum

C hcscQ

hcscdrt

C h z

s dct z

min 2h

s

2h

s

c c 0

…(20)

c c c c

z dr

t t t

If hc 0, then hc + sc > sc Z < Q

i.e. the order of level to which the inventory is raised is kept less than the total demand Q to create shortages.

C h z2 drt z

r

r

c s

c s

t 2d t2 c t

2

2

2

2

s d t z 2 s

Theorem 5 :

c r c 0

…(21)

The minimum cost of economic lot size under uniform rate of demand with finite rate of production and having shortages is given by

2dr t t

From (20) we have

z

h s s d 2h s s d

hc sc sc drt

2 c c c r c c c r

2hc sc

hc sc

z sc drt

…(22)

Proof : In orer to prove the theorem we will assume all the assumption of theorem (4) only considering production is instances.

Let q be the order quantity per production

hc sc

Using in the value z in (21) we have

hc scdrt sc d t scdrt

2d h s t r h s

run and z be the order level to which the inventory is raised at the begining of a run of

r c c

c c

time interval t

sc

d t

scdrt

s

c 0

\ Q = d t ….(18)

2d t2 r

h s t2

r r c c

or t2 2sc

hc sc

q* d t* d

2sc

sc hc

hcscdr

t r

hcscdr

t t*

2sc hc sc

q*

2dr sc hc sc

hcscdr

…(23)

hcsc

2C s h

which is the required economic lot size formula.

Now,

2C

z2

h z2

c c

drt

z

Cmin, which can be evaluated substantly the value of z from (22) on (19) we have

c sc

2 2

2

2 2

2

2

t drt

t

C z, t

hc scdrt

s 2d z

2z2 2s

2drt hc sc

c r

2

2

c

s s d t 2 s

2dr t

t3 t3

c d t c r c

2 r h s t

h z2

s z2

2s

drt

c c

c c c

h s2d t s pd t s

d t3 d t3 t3

c c r c c r c

r r

2h s 2 2h s 2 t

2C h z s z

c c c c

and

c c

h s2d t h

s s

h s d t s

t z d t2

d t 2

c c r c c

c c c r c

r r 2h

s 2

t 2h

s t

c c c c

2C

t2

2C

z2

2C 2

t z

have

substituting the value of t from (23) we

h z2

s z2

2s h s h z s z 2

C hcscdr

2sc

hc sc

c

c

c

c c

c c

min

r r r r r

r r r r r

d t3

d t3

t3 d t

d t 2

d t 2

2hc sc

hcscdr

2sc

sc hc

s1

hc sc dr

d t4

c 2s1 h

s

r c c c

Obviously for the value of t given by (23)

2 2 2 2 2 2

hcscsc dr

hcscsc dr

C 0, C 0 and C

C C 0

2h

s 2h

s

z 2

t 2

z 2

t 2

t z

c c

c c

C is minimum for the value of t given by

h s s d 2h s s d

(23) 2

c c c r

c c c r

The optimum order quantity for minimum

2hc sc

hc sc

cost is given by which is economic lot size formula.

Numerical Examples Example 1 :

A commodity is to supplied at a constant rate of 200 units perday supplies of any amount can be had at any required time but each ordering cost Rs. 50 cost of holding the commodity on inventory Rs.2 per unit per day while the delay in the supply of the item induces a penality of Rs.10 per unit delay of one day found the optimal policy (q, t) where t is the reorder cycle period end of the convention level

after recorder. What would be the best policy

Example 2:

A company has to purchase of a T.V. tube where demand dr = 2000 tubes per annum, = Rs. 150 per order and hc = Rs. 240 per item per annum, suppose now the supplier inform that if the order size at least 800 units he is prepared to supply the tubes at a discounted price of Rs.

9.80 per tube. Now we shall examine whether the offer of the supplier is attractive or not. Solution :

We have EOQ

of the penality cost becomes.

Solution :

dr = 200 cost per day

= Q*

2sc dr

hc

2 2000150 500 units.

2.40

2.40

2000

c

c

s1 = Rs. 50 per order

hc = Rs. 2 per unit per days

sc = Rs. 10 per unit per day

Total cost = 2000 Ã— 10 + 500 Ã— 150

500

+ 2 Ã— 2.40 = Rs. 21,200.00

q*

2drsc hc

hc sc sc

Now at a quantity Q = 800 to be purchase the price available is Rs. 9.80.

Total cost for Q for 800

2 50 200 2 10

2 10

= 2000 Ã— 9.80 +

2000

800 Ã—150 +

800

2 Ã—2.40

= 109.5 unit = 110 units

= Rs. 20,935 < Rs. 21,200

t

t

* Q*

dr

110 2 day (approximate)this

Hence it is go for a price discount to order for 800 units or more.

200

200

means that the optimum order quantity of 110 units must be supplied after 2days if the penality cost becomes

To ta l c o s t w ith o u t

Y pric e B re a k

P r ice B reak

Q*

2drsc

hc

250 200 100 units

B

2

2

A

To ta l c o s t w ith pric e B re a k

* Q* 100 1 O X

t days

D 200 2

200 800

Fig – 3

#### (Order size cost curve for price break model)

From the above figure we conclude that the price discount at a quantity of 800 units. At this stage the total cost is lower than the total cost corresponding to 500 units.

CONCLUSION :

In our work we hve tried to specify some basic theorems and examples like the correct amount and quality of items on the hand at the time required with a minimum expenditure of investment constituent with the business expendiency. It implies minimizing the inventories while at the same time ensuring that stockouts donot occur and production does not suffer. In our work we have made an attent to include increased costs arise from the extra stock holding cost caused by the average stock level being higher due to the larger order quantity.

References :

Bishop, J.L., (1974) : Experience with a successful system for forecasting and inventory control. Operations Research, 22(6), 1224 – 1231.

Prasad, S. (1994) : Classification of inventory models and system.

International Journals of production Economics, 34, 209- 22.

Zipkin, P. (1995) : The second golden age of Inventory management. Paper presented at the INFORMS, Los Angeles.

Spencer, M.S and J.F Cox (1995) : An analysis of the product – process Matrix and Repetitive manufacturing. International Journal of production Research, 33 (5), 1275 = 94.

Silver, E.A. (1981) : Operation research in inventory management. A review and cirtique. Opns. Res. 628-645.

Urgeleitti, T.G. (1983) : Inventory control models and problems, Eur. J. Opl Res 14 : 1 12.

Wagner, H.M. (1980) : Research portfolio for inventory management and production planning system. Opns Res. 28 : 225 475.

Ziegler, D (1996) : Selecting software for cycle counting APICS The performance Advantage, November 44 – 47.

Rosenblatt, M.J. and Lee, H.L. (1986) : Economic Production Cycles with imperfect production processes, IIE Trans. 14 : 48 – 55.

Kaplan, R.S. (1988); One cost system Isnt Enough, Havord Business Review (Jan.-Feb), 61 – 66.

Silver, E. D. pyke and R. Peterson (1998), Inventory management and production planning and Scheduling, 3rd ed. Wiley, New York.

Waters, C. (1992). Inventory Control and management, Wiley, New York.

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