Production Order Quantity Model

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Production OrderQuantity Model



Production Order Quantity

Model In EOQ Model, We

assumed that the

entire order wasreceived at one time.

However, Some

Business Firms mayreceive their ordersover a period of time.




Model Such cases require a different inventory

model.

Here, we take into account the dailyproduction rate and daily demand rate.




Model




Model Since this model is especially suitable

for production environments, It is

called Production Order Quantity Model .

Here, we use the same approach as weused in EOQ model.

Lets define the following:




Model p: Daily Production rate (units / day)

d: Daily demand rate (units / day)

t: Length of the production in days.

H: Annual holding cost per unit




Model Average Holding Cost = (Average

Inventory) . H

= (Max. Inventory / 2) . H




Model In the period of production (until the

end of each t period):

Max. Inventory = (Total Produced) – (Total Used)

= p.t - d.t




Model Here, Q is the total units that are

produced.

Therefore,

Q = p.t t = Q/p




Model If we replace the values of t in the Max.

Inventory formula:

Max. Inventory = p (Q/p) - d(Q/p) = Q - dQ/p = Q (1 – d/p)




Model Annual Holding Cost = (Max.

Inventory / 2) . H = Q/2 (1 –

d/p) . H

Annual Setup Cost = (D/Q) . S




Model Now we will set

Annual Holding Cost = AnnualSetup Cost

Q/2 (1 – d/p) . H = (D/Q) . S




Model




Model This formula gives us the optimum

production quantity for the Production

Order Quantity Model.

It is used when inventory is consumed

as it is produced .



Backorder Inventory

Model

In this model, we

assume that stock outs (andbackordering) are

allowed.



Backorder InventoryModel

In addition to previous assumptions, weassume that sales will not be lost due to

a stock out. Because, we will back order any

demand that can not be fulfilled.




B: Backordering cost per unit per year

b: The amount backordered at the time

the next order arrives

Q – b: Remaining units after thebackorder is satisfied




Total Annual Cost = Annual Setup Cost+ Annual Holding Cost + Annual

Backordering Cost Annual Setup (Ordering) Cost = (D/Q) . S

Annual Holding Cost = (Average

Inventory Level) . H




By using the graphical ratios, we knowthat:

T1 / T = (Q – b) / Q Therefore,if we replace T1/T in the above

equation we get

Average Inventory Level = (Q – b)2 / 2Q




By using the graphical ratios, we knowthat:

T2 / T = b / Q Therefore, if wereplace T2/T in the above equation weget

Average Backordering = b2 / 2Qand




We find optimum order quantity (Q*)and optimum backordering quantity

(b*) by taking the derivatives of dTC/dQ = 0 and dTC / db = 0 andthen putting the values in their places.



Quantity Discount Model

A quantity discount is simply a reducedprice (P) for an item when it is

purchased in LARGER quantities. A typical quantity discount schedule is

as follows:




Since the unit cost for the Thirddiscount is the lowest, We might be

tempted to order 2000 or more units. However, this quantity might not be the

one that minimizes the Total Cost.

Remember that, As the quantity goesup, the holding cost increases.




Here, there is a trade off betweenreduced product price (P) and increased

holding cost (H). Total Cost = Setup Cost + Holding

Cost + Product Price (Cost)

Total Cost = DS / Q + QH / 2 + PDwhere P is the price per unit




To determine the minimum Total Cost,we perform the following process which

includes 4 steps:




Step 1 : Assume that

I: is a percentage value, and

I . P represents the holding cost as apercentage of price per unit (P).




For each discount alternative, calculatea value of Q* = [2DS / IP]1/2

Here, instead of using a value of H, theholding cost is equal to I . P

That is, If the item is expensive (such

as a Class A Item), Its holding cost willbe higher.




Since the price of item (P) is a factor in Annual Holding Cost, we can no longer

assume that the holding cost isconstant (such as H) when pricechanges.




Step 2 : For any discount alternative,

If the calculated optimum order

quantity (Q*) is too low to qualify forthe discount range,

Then, Adjust the order quantity upward

to the lowest quantity that will qualifyfor the particular discount alternative.




Step 3 : Using the total cost (TC)equation above, compute a total cost

for every order quantity (Q). Use the adjusted Q values .




Step 4 : Select the discount alternativewhich has the minimum Total Cost

(TC).



Example

Consider the quantity discount schedulegiven in the beginning (above).

Assume that the Ordering (Setup) Cost(S) is $49 per each order.

Annual Demand (D) is 5000 units, and

Inventory carrying charge is apercentage (I=0.20) of product cost(P).



Example

Question : What order quantity willminimize the total inventory cost.

Answer :

Step 1 : Compute Q* for every discountrange.



Example



Example

Step 2 : Adjust values of Q* that arebelow allowable discount ranges.

- For Q1, allowable range is 0-999. SinceQ1* = 700 is between 0 and 999, Itdoes not have to be adjusted.



Example

- For Q2, allowable range is 1000-1999.Since Q2* = 714 is not in the allowed

range, we adjust it to the lowestallowable value, That is Q2* = 1000.

- For Q3, allowable range is 2000-. Since

Q3* = 718 is not in the allowed range,we adjust it to the lowest allowablevalue, That is Q3* = 2000.



Example

Step 3 : Compute total cost for each of the order quantities (Q*)



Example



Example

Step 4 : An Order quantity of 1000 unitswill minimize the total cost.

However, if the third discount cost islowered to $4.65, selecting Thisdiscount alternative (2000 units) would

be the optimum solution.



Probabilistic Models

So far we assumed that demand isconstant and uniform.

However, In Probabilistic models,demand is specified as a probabilitydistribution.

Uncertain demand raises the possibilityof a stock out (or shortage). (Why?)




One method of reducing stock outs is tohold extra inventory (called Safety

Stock ). In this case, we change the ROP

formula to include that safety stock

(ss).




ROP = d . L

d = daily demand, and

L = Order Lead Time

Now it will be as follows:

ROP = d . L + (ss)

where (ss) is the safety stock



Example

AMP Ltd. company determined its ROP= 50 units.

Its holding cost (H) is $5 per unit peryear.

Its stock out cost (B) is $40 per unit.

Probability of stock out is based on thefollowing probability distribution:



Example



Example

Question : Find the Level of Safety Stock (ss) that minimizes the total additional

holding cost and Stock out costs(annually).

Stock out cost is an expected cost; Thatis:



Example



Example

Possible stock outs per year is actually thenumber of orders per year (D/Q).

Since it is not known (or not given) assumethat it is 6 times / year.

For zero safety stock, there is no additionalholding cost for extra (safety) stock.

But there are stock out costs for two levels:



Example

1) If demand is 60 units at the ROP, Thena shortage of 10 units will occur.

(Because, ROP is 50 units)

2) If demand is 70 units at the ROP, Then

a shortage of 20 units will occur.



Example



Example

The safety stock with the lowest totalcost is (ss = 20) units.

Therefore, ROP = 50 + 20 = 70 units.

Fi di A S f t St k



Finding A Safety Stock Level

Managers may want

to limit thepossibility of stock out only to a smallpercentage, say 5%.

Fi di A S f t St k



Finding A Safety Stock Level

If demand level is assumed to be anormal distribution,

By using mean and standard deviationof the normal distribution,

We can determine a safety stock that is

necessary for %95 service level.



Example

The SAC company carries an inventoryitem that has a normally distributed

demand. The mean demand is (=350) units and

standard deviation is (=10).



Example



Example

We use the properties of a standardizednormal curve to get a z value that

corresponds to the.95 of the curve.

Using a standard Normal table we find z

= 1.65 for 95% confidence.



Example

Reorder point is elevated up by 16.5 units.



Example

Production Order Quantity Model

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