Inventry Control

7/27/2019 Inventry Control

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Inventory management or Inventory Control is one of the techniques of Materials

Management which helps the management to improve the productivity of capital by

reducing the material costs, preventing the large amounts of capital being locked up for

long periods, and improving the capital-turnover ratio. The techniques of inventory

control were evolved and developed during and after the Second World War and havehelped the more industrially developed countries to make spectacular progress in

improving their productivity.

Definition:

The word inventory means a physical stock of material or goods or commodities or

other economic resources that are stored or reserved or kept in stock or in hand for

smooth and efficient running of future affairs of an organization at the minimum

cost of funds or capital blocked in the form of materials or goods (Inventories).

The function of directing the movement of goods through the entire manufacturing

cycle from the requisitioning of raw materials to the inventory of finished goods in

an orderly manner to meet the objectives of maximum customer service with

minimum investment and efficient (low cost) plant operation is termed as inventory

control.

The inventory control problem consists of two basic factors:

1) When to order

2) How much to order

When to order

This is related to the lead time of an item. Lead time may be defined as the time interval

between the placement of order for an item & its receipt in stock. It may be

replenishment order on an outside firm or within the works. There should be enough

stock for each item so that customers orders can be reasonably met from this stock until

replenishment. This stock level, known as recorder level, has, therefore, to be determined

for each item. It is determined by balancing the cost of maintaining these stocks and thedisservices to the customer if his orders are not filled in time.

How much to order


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As already discussed, each order has associated with it the ordering cost or acquisition

cost. To keep it low, the number of orders should be as few as possible i.e. the order size

should be large. But large order size would imply high inventory cost. Thus the problem

of how much to order is solved by compromising between the acquisition costs and

inventory carrying costs.

The inventory control policy of an organization depends upon the demand characteristics.

The demand for an item may be independent or dependent. For instance, the demand for

the different models of television sets manufactured by a company does not depend upon

the demand of any other item, while the demand of its various components will depend

upon the demand of the television sets and may be arithmetically computed from the

latter. The independent demand is usually ascertained by extrapolating the past demand

history i.e. by forecasting. The order level can be fixed from the forecasts and the lead

time. Thus while in the case of dependent demand, simple arithmetic computations are

enough to ascertain requirement of the components, in the case of independent demand

items, statistical forecasting techniques have to be employed. The discussion of these

forecasting techniques will be taken up later in this chapter. The family tree drawn in the

next section gives an idea of the various inventory control policies.

Classification of Inventories

Inventories may be classified as those which play direct role during manufacture or

which can be identified on the product and the second one are those which are required

for manufacturing but not as a part of production or cannot be identified on the product.The first type is labeled as direct inventories and the second are labeled as indirect

inventories.

Further classification of direct and indirect inventories is as follows:

(A) Direct inventories

(i)Raw material inventories: The inventory of raw materials is the materials used in the

manufacture of product and can be identified on the product. In inventory control

manager can concentrate on the

(a) Bulk purchase of materials to save the investment,

(b) To meet the changes in production rate,

(c) To plan for buffer stock or safety stock to serve against the delay in delivery of

inventory against orders placed and also against seasonal fluctuations.


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(ii) Work-in -process inventories or in process inventories: These inventories are of

semi-finished type, which are accumulated between operations or facilities. As far as

possible, holding of materials between operations to be minimize if not avoided. This is

because; as we process the materials the economic value (added labour cost) and use

values are added to the raw material, which is drawn from stores. Hence if we hold thesesemi finished material for a long time the inventory carrying cost goes on increasing,

which is not advisable in inventory control. These inventories serve the following

purpose:

(a) Provide economical lot production,

(b) Cater to the variety of products,

(c) Replacement of wastages,

(d) To maintain uniform production even if sales varies.

(iii) Finished goods inventories: After finishing the production process and packing, the

finished products are stocked in stock room. These are known as finished goods

inventory. These are maintained to:

(a) To ensure the adequate supply to the customers,

(b) To allow stabilization of the production level and

(c

) To help sales promotion programme.

(iv)Spare parts inventories: Any product sold to the customer, will be subjected to wear

and tear due to usage and the customer has to replace the worn-out part. Hence the

manufacturers always calculate the life of the various components of his product and try

to supply the spare components to the market to help after sales service. The use of such

spare parts

(iv) Scrap or waste inventory: While processing the materials, we may come across

certain wastages and certain bad components (scrap), which are of no use. These may be

used by some other industries as raw material. These are to be collected and kept in aplace away from main stores and are disposed periodically by auctioning.


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Classification of Fixed Order Quantity Inventory Models

(B) Indirect Inventories

Inventories or materials like oils, grease, lubricants, cotton waste and such other materials

are required during the production process. But we cannot identify them on the product.

These are known as indirect inventories. In our discussion of inventories, in this chapter,

we only discuss about the direct inventories.

Inventories may also be classified depending on their nature of use. They are:

(i) Fluctuation Inventories: These inventories are carried out to safeguard the fluctuation

in demand, non-delivery of material in time due to extended lead-time. These are sometime called as Safety stock or reserves. In real world inventory situations, the material

may not be received in time as expected due to trouble in transport system or some times,

the demand for a certain material may increase unexpectedly. To safeguard such

situations, safety stocks are maintained. The level of this stock will fluctuate depending

on the demand and lead-time etc.


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(ii) Anticipation inventory: When there is an indication that the demand for companys

product is going to be increased in the coming season, a large stock of material is stored

in anticipation.

Some times in anticipation of raising prices, the material is stocked. Such inventories,

which are stocked in anticipation of raising demand or raising rises, are known as

anticipation inventories.

(iii)Lot size inventory or Cycle inventories: This situation happens in batch production

system.

In this system products are produced in economic batch quantities. It sometimes happens

that the materials are procured in quantities larger than the economic quantities to meet

the fluctuation in demand. In such cases the excess materials are stocked, which are

known as lot size or cycle inventories.

(iv) Transportation Inventories: When an item is ordered and purchased they are to be

received from the supplier, who is at a far of distance. The materials are shipped or

loaded to a transport vehicle and it will be in the vehicle until it is delivered to the

receiver. Similarly, when a finished product is sent to the customer by a transport vehicle

it cannot be used by the purchaser until he receives it. Such inventories, which are in

transit, are known as Transportation inventories.

(v)Decoupling inventories: These inventories are stocked in the manufacturing plant as a

precaution, in case the semi finished from one machine does not come to the nextmachine,

Inventory Models with Deterministic Demand

It is extremely difficult to formulate a single general inventory model which takes into

account all variation in real systems. In fact, even if such a model were developed, it may

not be analytically solvable. Thus inventory models are usually developed for some

specific situations.

In this section we shall deal with situations in which demand is assumed to be fixed andcompletely known. Models for such situation are called economic lot size models or

economic order quantity models.


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1. Model 1 (a) Classical EOQ Model( Demand Rate Uniform, Replenishment

Rate Infinite)

This is one of the simplest inventory models. A stockiest has order to supply

goods to customer at uniform rate R per unit time. Hence demand is fixed and

known. No shortages are allowed; consequently the cost of shortage, C2 is

infinity. He places an order with the manufacturer every t time units, where t is

fixed; and the ordering cost per order is C3. Replenishment time is negligible i.e.

replenishment rate is infinite so that replacement is instantaneous (lead time is

zero). The holding cost is assumed to be proportional to the amount of inventory

as well as the time inventory is held. This the cost of holding inventory I for time

T is C1IT1 where C1 is the cost of holding one unit in inventory for a unit of time.

The cost coefficients C1, C2 and C3 are assumed to be constants. The stockiest

problem is to determine

(a) How frequently he should place the order.

(b) How many units should be ordered in each order

This model is illustrated schematically in 12.1

If orders are paces at intervals t, a quantity q = Rt must be ordered in each order.

Since the stock in small time dt is Rtdt, the stock in time period t will be

=== qtRrdtRt 21

2

1.

2

1

0

Area of inventory triangle OAP

Cost of holding inventory during time t= C1 Rt2


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Ordering cost to place an order= C3

Total cost during time t= C1 Rt2 + C3

Average total cost per unit time, C(t) =t

CRtC 31

2

1+ (12.1)

C will be minimum if 0)(=

dt

tdCand 0

)(2

2

=dt

tCdis positive

Differentiating equation (12.1) twice w. r. t. t

3

3

2

2 2)(

t

C

dt

tCd= , which is positive for value of t given by the above equation

Thus, C(t) is minimum for optimal time interval.

RC

Ct

1

30

2=

Optimum quantity q0 to be ordered during each other

1

3

00

2

C

RCRtq ==

Which is known as the optimal lot size ( or economic order quantity) formula due to

R. H. Wilson. It is also called Wilsons or square root formula.

Any other order quantity will result in a higher cost.

The resulting minimum average cost per unit time.

3

13

1

310

2

2

2

1)(

C

RCC

RC

CRCqC +=

RCCRCCRCC 313131 2

2

1

2

1=+= (12.4)

Also the total minimum cost per unit time, including the cost of the time.

CRRCC += 312 (12.4a)


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Where, C is cost/ unit of the item.

Equation (12.1) can be written in an alternative form by replacing t by q/R as

q

RCqCqC 31

2

1)( += (12.5)

It may be realized that some of the assumptions made are not satisfied in actual practice.

For instance, it is seldom that a customer demand is known exactly and that

replenishment time is negligible.

Corollary 1: In the above model if the order cost is C3 + bq instead of being fixed,

where b is the order cost per unit item, we can prove that there is no change in the

optimum order quantity due to the changed order cost.

Proof: The average cost per unit time, )(2

1)( 31 bqC

q

RqCqC ++= [From eq.

(12.5)]

For the minimum cost 0)(=

dq

qdCand 0

)(2

2

=dq

qCdis positive

i.e. 02

12

31 =

q

RCC or

1

32

C

RCq =

3

2)( 32

2

q

RC

dq

qCd= , which is necessarily positive for above value of q.

1

30

2

C

RCq = , which is same as equation (12.3)

Hence there is no change in optimum order quantity as a result of change in the order

cost.


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Corollary 2: in model 1 (a) discussed above; the lead time has been assumed to be

zero. Most practical problems however have a positive lead time L from the time the

order for the item is places until it is actually delivered. The ordering policy of the

above model, therefore, must satisfy the reorder point.

If L is the lead time in days and R is the inventory consumption rate in units per day,

the total inventory requirements during the lead time = LR. Thus we should place and

order q as soon as the stock level becomes LR. This is called reorder point p = LR.

In practice, this is equivalent to continuously observing the level of inventory until the

reorder point is obtained. That is why the economic lot size model is also called

continuous review model.

If a buffer stock B is to be maintained, recorder point will be

LRBp += (12.6)

Furthermore, if D days are required for reviewing the system,

++=++=

22

DLRB

RDLRBp (12.7)

Assumptions in EOQ Formula


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Following simplifying assumptions have been made while deriving the economic

order quantity formula:

1. Demand is known and uniform (constant).

2. Shortages are not permitted; as soon as the stock level becomes zero, it isinstantaneously replenished.

3. Replenishment of stock is instantaneous or replenishment rate is infinite.

4. Lead time is zero. The moment the order is places, the quantity ordered is

received.

5. Inventory carrying cost and ordering cost/order remain constant over time. The

former is linearly related to the quantity ordered and the latter to the number of

orders.

6. Cost of the item remains constant over time. There are no price-breaks or quantity

discounts.

Limitations of EOQ Formula

The EOQ formula has a number of limitations. It has been highly controversial since a

number of objections have been raised regarding its validity. Some of them are

1. In practice the demand is neither known with certainty not it is uniform. If the

fluctuations are mild, the formula can be applicable but for large fluctuations it

loses its validity. Dynamic EOQ models, instead, may have to be applied.

2. The ordering cost is difficult to measure. Also it may not be linearly related to the

number of orders as assumed in the derivation of the model. The inventory

carrying rate is still more difficult to measure and even to define precisely.

3. It is difficult to predict the demand. Present demand may be quite different from

the past history. Hardly any prediction is possible for a new product to be

introduced in the market.

4. The EOQ model assumes instantaneous replenishment of the entire quantity

ordered. In practice, the total quantity may be supplied in parts. EOQ model is not

applicable in such a situation.

5. Lead time may not be zero unless the supplier is next-door and has sufficient stock

of the item, which is rarely so.


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6. Price variations, quantity discounts and shortages may further invalidate the use of

the EOQ formula.

However, the flatness of the total cost curve around the minimum is an answer to many

objections. Even if we deviate from EOQ within reasonable limits, there is no substantial

change in cost. For example, if because of inaccuracies and errors, we have selected an

order quantity 20% more (or less) than q0 the increase in total cost will be less than 2%.

Model 1 (b) ( Demand Rate Non-Uniform, Replenishment Rate Infinite)

In this model all assumptions are same as in model 1(a) with the exception that instead of

uniform demand rate R, we are given some total demand D, to be satisfied during some

long time period T. Thus demand rates are different order cycles.

Let q be the fixed quantity ordered each time the order is placed.

Number of orders,q

DN =

If t1 is the time interval between orders 1 & 2, t 2 is the time interval between orders 2 & 3

and so on, the total time T will be

= t1 + t2 + + tn

This model is illustrated schematically in figure 12.3


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Holding costs for time period T will be

11211 )2

1......()

2

1()

2

1( CqtCqtCqt n+=

TqCtttqC n 12112

1).....(.

2

1=+=

And the ordering cost will be

=C3 . N

q

DC .3= , where C3 is the ordering cost per order.

Total cost equation for fixed order size q will be

q

DCTqCqC 31

2

1)( += (12.9)

For minimum cost, [ ] 0)( =qCdq

dand [ ])(2

2

qCdq

dshould be positive.

Differentiating equation (12.9) w.r.t. q,


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02

1)]([

231==

q

DCTCqC

dq

d

1

3

1

3 /.22

C

TDC

TC

DCq ==

And 23

2

2 2)]([

q

DCqC

dq

d= , which is positive for the value of q given above

Optimal lot size,1

30

)/(2

C

TDCq = , (12.10)

And minimum total cost,

)/(2

)/(2..

2

1)(

3

13

1

310

TDC

CDC

C

TDCTCqC +=

= )/(2 31 TDCC (12.11)

From equations (12.10) and (12.11) we find that results for this model can be obtained if

the uniform demand rate R in model 1 (a) is replaced by average demand rate D/T.

Model 1 (C) ( Demand Rate Uniform, Replenishment Rate Infinite)

In the classical EOQ model the replenishment the rate was assumed to be infinite; the

entire quantity ordered was delivered in a single lot. This is possible only for bought-out

items and is simple unthinkable for made in items. Such items are produced by the

production department of the organization at a constant rate and are also supplied to the

customers at a constant rate. When the production starts, a fixed number for units are

supposed to be added to inventory each day till the production run is completed;

simultaneously, the items will be demanded at a constant rate, as stipulated earlier.

Obviously, the rate at which they are produced has to be higher than the consumption

rate, for only then can there be the built-up of inventory.

It is assumed that run sizes are constant and that a new run will be started whenever

inventory is zero. Let

R = number of items required per unit time,


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K = number of produced per unit time

C1 = cost of holding per tem per unit time.

C3 = cost setting up a production run,

q = number of items produced per run,

t = interval between runs

Figure 12.4 shows the variation of inventory with time.

Here, each production run of length t consist of two parts t1 and t2 where

(i) t1 is the time during which the stock is building up at a constant rate of K R

units per unit time,

(ii) t2 is the time during which there is no production (or supply or replenishment)

and

inventory is decreasing at a constant demand rate R per unit time

Let Im be the maximum inventory available at the end of time t 1 which is expected to be

consumed during the remaining period at the demand rate R

Then Im =(K-R)t1 orRK

It m

=

1 ( 12)


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Now the total quantity produced during time t1 is q and the quantity consumed during the

same period is therefore the remaining quantity available at the end of time t1 is

Im =q-Rt1

RKRIq m

=

qRK

RIm =

+1 or q

K

RKIm

=

Now holding cost per production run i.e. for time period t

1...2

1CtIm=

And setup cost per production run = C3

Total average cost per unit time,

tCCItIC mm /.2

1)( 31, += (15)

Or tCCqK

RKtqC /..

2

1),( 31 +

=

Or Rq

CCq

K

RKqC

/..

2

1)( 3

1 +

= Rtq =

q

RCqC

K

RK 31.

)(.

2

1+

= (16)

For minimum value of C(q),

02

1)]([

2

31 =

=

q

RCC

K

RKqC

dq

d, which gives

1

3

1

3

1

3 2....2)(

2CRC

RKK

RKRK

CC

CRKRKCq

===

And 33

2

2 2)]([

q

RCqC

dq

d= , which is positive


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Optimal lot size,RK

RK

C

Cq

= .

2

1

3

0 , (17)

Model 2 (a) (Demand Rate Uniform, Replenishment Rate Infinite, Shortages

Allowed)

This model is just the extension of model1 (a), allowing shortages. Let

R = number of the items required per unit time i.e. demand rate,

C1 = cost of holding the item per unit time,

C2 = shortage cost per item per unit time,

C3 = ordering cost/order,

q = number of items ordered in one order,

q = Rt,

t = interval between orders,

Im = number of items that form inventory at the beginning of time interval t. lead time is

assumed to be zero. Fig. 12.5 shows the variation of inventory with time.


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Here the total time period T is divided into n equal time intervals, each of value t. the

time interval t is further divided into two parts t1 and t2.

i. e. t = t1 + t2

Where t1 is the time interval during which items are drawn from inventory ant t2 is the

interval during which the items are not filled. Using the relationship of similar triangles,

Model 2 (b) (Demand Rate Uniform, Replenishment Rate Infinite, Shortages

Allowed, Time Interval Fixed)

Time interval t is fixed which means that inventory is to be replenished after every fixed

time t. All other assumptions of model 2 (a) hold good.


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Model 2 ( C ) (Demand Rate Uniform, Production Rate Finite, Shortages Allowed)

This model has the same assumptions as in model 2 (a) except that production rate is

finite. Figure shows the variation of inventory with time.

Referring to figure we find that inventory is zero in the beginning. It increases at constant

rate (K R) for time t1 until it reaches a level Im. There is no replenishment during timet2, inventory decreases at constant rate R till it becomes zero. Shortage starts piling up at

constant rate R during time t3 until this backlog reaches a level s. lastly, production starts

and backlog is filled at a constant rate K R during time t4 till the backlog becomes zero.

This completes one cycle; the total time taken during this cycle is


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This cycle repeats itself over and over again.

Now holding cost during time interval t

Now C is a function of six variables Im, s, t1, t2, t3 and t4 but we can derive relationships

which determine the values of Im, t1, t2, t3 and t4 in terms of only two variables q and s.

An inventory policy is given when we know how much to produce i. e. q and when tostart production, which can be found if s is known.

The manufacturing rate multiplied by the manufacturing time gives the manufactured

quantity.

INVENTORY MODELS WITH PROBABILISTIC DEMAND

The models discussed in previous sections are only artificial since in practical situations,

demand is hardly known precisely. In most situations demand is probabilistic since only

probability distribution of future demand, rather than the exact value of demand itself, is

known. The probability distribution of future demand is usually dete4mined from the date

collected from past experience. In such situations we choose policies that minimize the

expected costs rather than the actual costs. Expected costs are obtained by multiplying the

actual costs for a particular situation with the probability or occurrence of that situation

and then either summing or integrating according as the probability distribution is

discrete or continuous.

Model 3 (a) (Instantaneous Demand, Setup Cost Zero, Stock Levels Discrete and

Lead Time Zero)


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Let R = discrete demand rate with probability PR,

Im = discrete stock level for time interval t,

t = constant interval between orders,

C1 = over-stocking cost (over-ordering cost). This is opportunity loss

associated with each unit left unsold

= C + Ch V,

C2 = under-stocking cost (under-ordering cost). This is opportunity loss

due to not meeting the demand.

= S-C-Ch/2 + Cs,

Where C is the unit cost price, Ch the unit carrying cost, Cs the unit shortage cost, S the

unit selling price and V is the salvage value. If value of any parameter is not given, it is

taken as zero.

Production is assumed to be instantaneous and lead time negligibly small. The problem isto determine the optimal inventory level Im, where R Im at the beginning of

each time interval. The variation of inventory with time for these two cases is shown in

fig. 12.7(a) and (b)

When R Im, as shown in

fig. 12.7(b) shortages occur.


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Then the optimal order quantity Im0 is determined when value of cumulative probability

distribution exceeds the ratio21

2

CC

C

+by computing

mm IRIR pCC

Cp


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When R Im as shown in fig.

shortages occur.

It can be shown that for optimum stock level,

Model 4 (b) (Continuous Demand, Setup Cost Zero, Continuous Stock Levels, Lead

Time Zero)

In this model, all conditions are same as in model 4(a) except that the stock levels are

continuous (rather than discrete). Therefore, probability f(R) dR will be used instead of

PRwhere f(R) is the probability density function of the demand rate R.

Then the optimal order Im0 is given by

INVENTORY MODELS WITH PRICE BREAKS


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In the inventory model discussed so far the production or purchase cost per unit was

assumed tpo be constant. It was not considered during their formulation since it did not

affect the level of inventory. In this section we shall consider a class of inventory

problems in which this cost is variable and depends upon the quantity manufactured or

purchased. This usually happens when discounts are offered for the purchase of largequantities. These discounts take the form of price breaks. For example, the price breaks

may be given as

Re. 1 per item for purchase of items upto 500,

Re. 0.95 per item for purchase of items upto 1,000,

Re. 0.90 per item for purchase of items 1,001 or more

Clearly, the purchase cost C(q) is a variable and is given by the expression

Such a variable cost must be considered in the inventory model. Further, as this variable

production or purchase cost per unit is more appropriate for purchased parts (because of

quantity discounts), we shall, hereafter, refer only to purchased parts and the problem,

then, is to determine

(i) how often the parts be purchased,

(ii) how many units should be purchased at any one time.

Inventory Models with One Price Break

To understand how the optimal order quantity can be determined in such a case, let us

consider example.

EXAMPLE

An automobile manufacturer purchases 2,400 castings over a period of 360 days.This requirement is fixed and known. These castings are subject to quantity discounts.

Ordering cost is Rs. 70,000/order and storage cost per day is 0.12% of the unit cost.

Determine the optimal purchase quantity if the supplier has offered the following unit

prices for the castings:

Unit price = Rs. 1,000 for q < 1,000


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= Rs. 950 for q >= 1,000

Solution

First we calculate the EOQ if the unit price of castings is Rs 950.

Therefore, the manufacturer should place order for 905 units. This, however, will

not be acceptable to the supplier since he will charge Rs. 950/unit only if the order is

placed for 1,000 or more units in an order. Therefore, this is an infeasible solution. To

avail unit price of Rs. 950, the manufacturer must place an order of at least 1,000 units.

Total cost per day for order quantity of 1,000 units

Since the earlier cost is lower, the optimal order quantity is 1,000 units. The total cost

curve (which is a stepped curve) is shown in Fig 12.9


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Cleary, the curve shows a marked drop in the total cost due to price discount at a quantity

of 1,000 units. At this level, the total cost is lower than the total cost corresponding to

882 units.

Inventory Models with Multiple Price Breaks

Sometimes the supplier may offer more than one price breaks. The solution procedure for

such problems is the extension of that used for single price break.

We first determine the EOQ with the lowest cost price. If it is feasible i.e. if the value

falls in the slab of the lot size for this cost price, then this is the optimal order quantity. If

it is infeasible we determine EOQ with the next lowest cost price. If this too is infeasible,

the next lowest cost is tried. The procedure is continued till the feasible EOQ is obtained.

The total cost corresponding to this quantity level is then calculated. Also, the total cost

of order quantities corresponding to cutoff points of the subsequent cost prices is

calculated. The quantity corresponding to the minimum of these total costs is the optimal

order quantity.

EXAMPLE


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MULTI-ITEM DETERMINISTIC MODEL

This model considers the inventory system consisting of several items. There are

limitations on production facilities or storage capacity or time or money. This limitationresults in an interaction between the different items so that it is not possible to consider

each item separately. However, simple cases can be handled by using the technique of

Lagrange multipliers.

Consider an inventory consisting of n items. For simplicity let us assume that production

is instantaneous, there is no quantity discount and that no shortages are permitted. Further

let us assume that the demand is known and uniform at a rate of Ri per unit time for the

ith item. Let C1i be the inventory holding cost per unit per unit time and C 3i be the setup

cost per production run for the ith item.

Total cost ofith item per production run

Limitation on Inventories

If now there is a limitation on inventories that restricts the average number of all types of

stocked items toIm the cost C (q1, q2, ., qn) must be minimized subject to the restriction

that

Notice that qi0 is dependent on o the optimal value of . Also for o = 0, qi

0 gives the

solution of the unconstrained case. The value of o can be found by systematic trial and

error. By definition < 0 for the above minimization case. Thus if we try successive

negative values of , its optimum value o should result in simultaneous values of qi0


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which satisfy the given constraint in equality sense. Thus determination of 0

automatically yields qi0.

Limitation on Storage Area

Let A be the maximum storage area available for n items and a i be the storage arearequired by one unit of ith item. If qi is the older quantity for ith item, the storage

requirements constraint becomes

Inventry Control

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