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Production OrderQuantity Model
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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.
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Production Order Quantity
Model Such cases require a different inventory
model.
Here, we take into account the dailyproduction rate and daily demand rate.
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Production Order Quantity
Model
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Production Order Quantity
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:
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Production Order Quantity
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
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Production Order Quantity
Model Average Holding Cost = (Average
Inventory) . H
= (Max. Inventory / 2) . H
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Production Order Quantity
Model In the period of production (until the
end of each t period):
Max. Inventory = (Total Produced) – (Total Used)
= p.t - d.t
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Production Order Quantity
Model Here, Q is the total units that are
produced.
Therefore,
Q = p.t t = Q/p
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Production Order Quantity
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)
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Production Order Quantity
Model Annual Holding Cost = (Max.
Inventory / 2) . H = Q/2 (1 –
d/p) . H
Annual Setup Cost = (D/Q) . S
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Production Order Quantity
Model Now we will set
Annual Holding Cost = AnnualSetup Cost
Q/2 (1 – d/p) . H = (D/Q) . S
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Production Order Quantity
Model
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Production Order Quantity
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 .
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Backorder Inventory
Model
In this model, we
assume that stock outs (andbackordering) are
allowed.
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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.
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Backorder InventoryModel
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
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Backorder InventoryModel
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Backorder InventoryModel
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
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Backorder InventoryModel
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Backorder InventoryModel
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
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Backorder InventoryModel
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Backorder InventoryModel
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
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Backorder InventoryModel
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Backorder InventoryModel
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.
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Backorder InventoryModel
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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:
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Quantity Discount Model
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Quantity Discount Model
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.
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Quantity Discount Model
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
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Quantity Discount Model
To determine the minimum Total Cost,we perform the following process which
includes 4 steps:
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Quantity Discount Model
Step 1 : Assume that
I: is a percentage value, and
I . P represents the holding cost as apercentage of price per unit (P).
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Quantity Discount Model
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.
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Quantity Discount Model
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.
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Quantity Discount Model
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.
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Quantity Discount Model
Step 3 : Using the total cost (TC)equation above, compute a total cost
for every order quantity (Q). Use the adjusted Q values .
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Quantity Discount Model
Step 4 : Select the discount alternativewhich has the minimum Total Cost
(TC).
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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).
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Example
Question : What order quantity willminimize the total inventory cost.
Answer :
Step 1 : Compute Q* for every discountrange.
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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.
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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.
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Example
Step 3 : Compute total cost for each of the order quantities (Q*)
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Example
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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.
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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?)
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Probabilistic Models
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).
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Probabilistic Models
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
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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:
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Example
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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:
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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:
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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.
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Example
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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
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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
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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.
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Example
The SAC company carries an inventoryitem that has a normally distributed
demand. The mean demand is (=350) units and
standard deviation is (=10).
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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.
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Example
Reorder point is elevated up by 16.5 units.
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Example