8 Inventory Management

Inventory Management

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Why do we have inventories?

Inventory is one of the most expensive assets of many companies.

It represents as much as 40% of total invested capital.

Inventory is any stored resource that is used to satisfy a current or future need.

Raw materials, work-in-process, and finished goods are examples of inventory.

Two basic questions in inventory management are (1) how much to order (or produce), and (2) when to order (or produce).

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Basic Functions of Inventory

Basic Functions of Inventory are:1. If product demand is high in summer, a firm

might produce during winter (Decoupling).

2. Inventory can be a hedge against price changes and inflation.

3. Another use of inventory is to take advantage of quantity discounts (when buying).

(Many suppliers offer discounts for large orders)

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ABC Analysis

ABC analysis divides on-hand inventory into three classifications on the basis of dollar (TL) volume.

It is also known as Pareto analysis. (which is named after principles dictated by Pareto).

The idea is to focus resources on the critical few and not on the trivial many.

(Annual Dollar Volume of an Item) = (Its Annual Demand) x (Its Cost per unit)

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ABC Analysis…

Class A items are those on which the annual dollar volume is high. They represent 70-80% of total inventory costs, but

they account for only 15% of total inventory items. Class B items are those on which annual dollar

volume is medium. They represent 15-25% of total dollar value, and they

account for 30% of total inventory items on the average.

Class C items are low dollar volume items. They represent only the 5% of total dollar volume, but

they include as many as 50-60% of total inventory items.

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ABC Analysis…

Percent of Annual Dollar Volume

Percent of Inventory Items

80

ClassA

Items

20

20

50 100

ClassC Items

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ABC Analysis…

Some of the Inventory Management Policies that may be based on ABC analysis include:

a) Class A items should have tighter inventory control.

b) Class A items may be stored in a more secure area.

c) Forecasting Class A items may warrant more care.

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Cycle Counting of Inventory

Inventory records must be verified through a continuing audit.

Such audits are known as (periodical) cycle counting.. (e.g., counting items at supermarket).

Cycle counting uses inventory classifications developed by ABC analysis. That is: Class A items are counted frequently, perhaps once a

month. Class B items are counted less frequently, perhaps

once a quarter. Class C items are counted perhaps once every six

months.

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Just-in-Time Inventory

Just in Time Inventory is the minimum inventory that is necessary to keep a system perfectly running.

With just in time (JIT) inventory, The exact amount of items arrive at the moment they are needed, Not a minute before OR not a minute after.

To achieve JIT inventory, Managers should Reduce the Variability Caused by some Internal and External Factors. (Goldratt’s boys scout example – Apply the pace of the

slowest boy). Existence of Inventory hides the variability. What causes variability?

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What causes variability?

Most variability is caused by tolerating waste (inventory). (1) For example, employees or machines

produce units that do not conform to standards. These are waste. And they cause variability.

(2) Or, engineering drawings are inaccurate, Again resulting in loss of production And consecutively resulting in Variability.

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What causes variability?...

These are the internal (controllable) factors that cause Variability.

However, Some of the variability is caused by some external factors.

For example, customer demands may change due to some external factors (such as competitors’ actions or promotions)

In summary, To achieve JIT inventory, Managers must begin with Reducing Inventory.

Reducing Inventory uncovers the Rocks located along the way on a river, And the water stream becomes more clear.

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Rocks located along the way

Others Waiting Time

Move Time

Queue Time

Set-up Time

Run Time

Input Output

Lead Time

Cycle Time

The section called “Others” are the Rocks on the river.

Those rocks include Quality Variability, In-transit Delays, Machine Breakdowns, Large Lot-sizes, Inaccurate drawings, Employee attendance variability.

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Just-In-Time Production

JIT production means (1) Elimination of Waste, (2) Synchronized Manufacturing, and (3) Little Inventory.

Reducing the order batch size can be a major help in reducing inventory.

Average Inventory = (Maximum Inventory + Minimum Inventory) / 2

Average Inventory drops as the inventory re-order quantity drops because the maximum inventory level drops.

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Kanban System

Moreover, the smaller the lot size, the fewer the problems are hidden.

One way to achieve small lot sizes is to Move Inventory through the shop Only as needed.

This is called a “pull” system. In this system, Ideal Lot size is 1.

Japanese call this system as “Kanban” system. Kanban is a Japanese word for Card. A card is used to signal the need for material in a

work center.

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Kanban System…

Sending a card authorizes the previous work center to send its finished batch to the subsequent work center.

Batches are typically very small. Such a system requires tight schedules and frequent set-ups for machines.

On the other hand, Small batches allow a very limited amount of faulty material, less damages, less space occupation, less material handling, less accidents, etc.

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Inventory Related Costs:Holding, Ordering and Set-up Costs Holding Costs are the costs associated with holding

or “carrying” inventory over time. It includes costs related to storage; such as

insurance, extra staffing, interest, and so on. Some example holding costs are:

building rent or depreciation, building operating cost, taxes on building, insurance on building, material handling equipment leasing or depreciation, equipment operating cost, handling manpower cost, taxes on inventory, insurance, etc.

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Inventory Related Costs:Ordering and Set-up Costs… Ordering Costs include, cost of supplies, order

processing, clerical cost, etc. The ordering cost is valid if the products are purchased

NOT produced internally. Set-up cost is the cost to prepare a machine for

manufacturing an order. Set-up cost is highly correlated with set-up time. Machines that traditionally have taken long hours to set

up Are Now being set up in less than a minute by employing FMSs or CIM systems.

Reducing set up times is an excellent way to Reduce Inventory.

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Inventory Models

Demand for an item is either dependent on the demand for other items or it is independent.

For example, demand for refrigerator is independent of the demand for cars.

But, demand for auto tires is certainly dependent on the demand of cars.

We will deal with the Independent Demand Situation.

In the dependent demand situation we use Material Requirement Planning (MRP) systems.

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What to answer?

In the independent demand situation, we should be interested in answering:

a) When to place an order for an item, and b) How much of an item to order.

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Independent Demand Inventory Models

There are Four Basic Independent Demand Inventory Models:

1) Economic Order Quantity (EOP) Model (the most known model).

2) Production Order Quantity Model.

3) Back order inventory model.

4) Quantity discount model.

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Economic Order Quantity (EOQ) Model EOQ model makes a number of assumptions:

1-) Demand is known and constant.2-) Lead time (the time between placement of order and

receipt of the order) is constant and known.3-) Orders arrive in one batch at a time, and they arrive in

one point in time.4-) Quantity discounts are not possible.5-) The costs include only setup cost (or ordering cost

when buying) and holding cost.6-) Orders are always placed at the right times.

Therefore, stock outs (or shortages) can be completely avoided.

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With these assumptions in EOQ Model, the graphic of inventory usage over time is as follows:

Inventory Level

Time

Q

AverageInventory Level

Usage Rate

0

Q = order quantity (That is also equal to the Maximum Inventory)

Minimum Inventory = 0

When inventory level reaches 0, a new order is placed and received.

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Objective: Minimize total cost

The objective of inventory models is to minimize total cost.

If we minimize the setup and holding costs, we will be able to minimize total cost.

As the quantity ordered (Q) increases, holding cost increases, and setup cost decreases.

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Where the total cost is minimum?

Annual Cost

Order Quantity (Q)

Annual Holding Cost

Setup Cost (Ordering Cost)

Total Cost

Minimum TotalCost

Optimal Order Quantity (Q*)

Optimal order quantity (Q*) occurs at a point where setup cost is equal to the total (annual) holding cost.

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Finding the optimum…

Optimal order quantity (Q*) occurs at a point where setup cost is equal to the total (annual) holding cost.

By using this fact, we can write an equation for Q* as follows:

D: Annual Demand in units for the inventory item.

S: Setup cost (or the ordering cost) for each order.Notice: (Setup cost for production, order cost for buying).

H: Annual Holding cost of inventory per unit.

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There will be (D/Q) times of ordering in a whole year.

Therefore, Annual Setup Cost = (D/Q) . S

Average Annual Holding Cost = (Average Inventory) . H = (Q/2) . H

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We get optimum point by setting:

Annual Setup Cost = Annual Holding Cost

(D/Q) . S = (Q/2) . H

Therefore,

Q2 = 2DS / H

Q* = [2DS / H]1/2

Q* value is also called as the EOQ.

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Example

An Inventory model has the following characteristics:

Annual Demand (D) = 1000 units

Ordering (Setup) cost (S)= $10 per order;

Holding cost per unit per year (H) = $.50

Assume that there are 270 working days in a year (excluding holidays and weekends).

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Example…

Questions: a) Find the Economic Order Quantity (Q*) for

this inventory model.

b) How many orders should be placed during one year?

c) What is the expected time between two consecutive orders?

d) What is the total annual cost of this inventory model?

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Example…

Answers:

a) EOQ= Q* = [2(1000)10 / .50]1/2 = 200 units

b) Expected number of orders placed during the year (N) = D / Q* = 1000 / 200 = 5 times.

c) Expected time between orders (T) = (Working days in a year) / N = 270 / 5 = 54 days.

d) Total Annual Cost = Annual Setup Cost + Annual Holding Cost

= DS / Q* + (Q*) H / 2

= 1000 (10) / 200 + (200) (.50) / 2= $100

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Proof of Optimality by Using Derivation

If we take the derivative of Total Cost (TC) function, based on the order quantity (Q), we get the following:

TC = DS / Q + (Q)H / 2

dTC/dQ = (- DS / Q2) + (H / 2)

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Proof of Optimality by Using Derivation…As a mathematic rule, if we set this derived

equation equal to zero, we get the optimal (minimum) point of the total cost function:

Therefore,dTC/dQ = (- DS / Q2) + (H / 2) = 0

From here, (- DS / Q2) = - H / 2 Q2 = 2DS / H

Q* = [2DS / H]1/2

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Proof of Optimality by Using Derivation… One more check is needed for the optimality

of Q. That is we take the second derivative of the

total cost function based on Q. If the second derivative is positive, the Q*

value is a real optimum. (Rule) In fact, second derivative is equal to 2DS / Q3

which is a positive value (It is a real optimum)

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Considering the Reorder Point

So far, we only decided how much to order (That is Q*).

Now, we should find what time to order. We assumed that firm will wait until its

inventory reaches to zero before placing an order.

And, we also assumed that the Orders will receive immediately.

However, there is a time between placement and receipt of an order.

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Considering the Reorder Point…

This is called LEAD TIME or delivery time. Here, we will use the term “Reorder Point”

(ROP) for when to order. ROP (in units) = (Demand Per Day) x (Lead

time for a new order in days)

ROP = d x L

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Reorder Point…

Inventory

Time (Days)

Q* Slope = d (units/day)

ROP (units)

L = Lead Time

ROP = d . L

When the inventory level reaches the ROP, a new order is required.

It will take a time that is equal to the Lead Time (L) to receive the new order.

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Reorder Point…

Here, Demand per day (d) is found by the following equation: d = D / Number of working days in a

year

This ROP equation assumes that demand is uniform and constant.

If this is not the case, an extra (safety) stock is added (because of uncertainty).

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Example

Annual demand for an item is D = 8000/year.

This year there will be 200 working days in a year.

Delivery of an order for this item takes 3 working days (L = 3 days).

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Example…

Questions:

a) Find the demand per day for this item.

b) What is the ROP for this item?

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Example…

Answers: a) Demand per day for this item (d) = 8000 / 200

= 40 units / day.

b) ROP = d . L = 40 . 3 = 120 units. When inventory level becomes 120 units, an

Order should be placed.

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Production Order Quantity Model

In EOQ Model, We assumed that the entire order was received at one time.

However, Some Business Firms may receive their orders over a period of time.

Such cases require a different inventory model.

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

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Production Order Quantity Model..

Inventory

Time

Maximum Inventory

t

Production occurs at a rate of p Demand

occurs at a rate of d

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Since this model is especially suitable for production environments, It is called Production Order Quantity Model.

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

Lets define the following: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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Average Holding Cost = (Average Inventory) . H

= (Max. Inventory / 2) . H

In the period of production:

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

= p x t – d x t

Here, Q is the total units that are produced.

Therefore,

Q = p x t and t = Q / p

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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)

Annual Holding Cost = (Max. Inventory / 2) . H = Q/2 (1 – d/p) . H

Annual Setup Cost = (D/Q) . S

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Now we will set:

Annual Holding Cost = Annual Setup Cost

Q/2 (1 – d/p) . H = (D/Q) . S

p

dH

DSQ

1

22

p

dH

DSQ

1

2*

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 (and backordering) are allowed.

In addition to previous assumptions, we assume 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 the backorder is satisfied

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Backorder Inventory Model…

Inventory

Time

(Q - b)

b

T1 T2

b

Q

0

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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

Average Average inventory Proportion of time Inventory = level during there is

inventory Level in stock period in stock

= (Q – b) / 2 . T1 / T

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By using the graphical ratios, we know that:T1 / T = (Q – b) / QTherefore, if we replace T1/T in the above

equation we getAverage Inventory Level = (Q – b)2 / 2Q

(Q – b)2

Annual Holding Cost = ---------- . H 2Q

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Annual Backordering Cost = (Average Backordering) . B

Average Backordering = (Average number of stock outs during out of stock period) x (Proportion of time inventory is on backorder)

Average Backordering = b / 2 . T2 / T

By using the graphical ratios, we know that:T2 / T = b / Q Therefore, if we replace T2/T in the above

equation we get:Average Backordering = b2 / 2Q and

b2

Annual Backordering Cost = ---------- . B 2Q

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DS (Q – b)2 b2 Total Cost (TC) = ------- + ---------- . H + --------- . B

Q 2Q 2QWe find optimum order quantity (Q*) and optimum

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

We find that:

B

BH

H

DSQ

2*

HB

HQb

*

*

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Quantity Discount Model

A quantity discount is simply a reduced price (P) for an item when it is purchased in LARGER quantities.

A typical quantity discount schedule is as follows:

Alternative Quantity Discount (%) Discount (unit) Price 1 0-999 0 $5.00 2 1000-1999 4 $4.80 3 2000- 5 $4.75

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Quantity Discount Model…

Since the unit cost for the Third discount 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 goes up, the holding cost increases.

Here, there is a trade off between reduced product price (P) and increased holding cost (H).

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Total Cost = Setup Cost + Holding Cost + Product Price (Cost)

Total Cost = DS / Q + QH / 2 + PD

where 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 a percentage of price per unit (P).

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

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

That is, If the item is expensive (such as a Class A Item), Its holding cost will be higher.

Since the price of item (P) is a factor in Annual Holding Cost, we can no longer assume that the holding cost is constant (such as H) when price changes.

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Step 2: For any discount alternative, If the calculated optimum order quantity (Q*) is too

low to qualify for the discount range, Then, Adjust the order quantity upward to the lowest

quantity that will qualify for 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 alternative which has the minimum Total Cost (TC).

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Example

Consider the quantity discount schedule given 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 a percentage

(I=0.20) of product cost (P). Question: What order quantity will minimize

the total inventory cost.

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Example…

Answer:

Step 1: Compute Q* for every discount range.

orderunitsIP

DSQ /700

)5)(2(.

)49)(5000(22*1

orderunitsIP

DSQ /714

)8.4)(2(.

)49)(5000(22*2

orderunitsIP

DSQ /718

)75.4)(2(.

)49)(5000(22*3

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Example…

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

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

- For Q2, allowable range is 1000-1999. Since Q2* = 714 is not in the allowed range, we adjust it to the lowest allowable 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 allowable value, That is Q3* = 2000.

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Example…

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

5000 x Unit price$49 x (5000 / 700) (Average Inventory x Holding Cost) =

(Q / 2) x (I x P) =

(700 / 2) x (5 x 0,20)

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Example…

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

However, if the third discount cost is lowered to $4.65, selecting this discount alternative (2000 units) would be the optimum solution.

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Probabilistic Models

So far we assumed that demand is constant and uniform.

However, In Probabilistic models, demand is specified as a probability distribution.

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

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Probabilistic Models

One method of reducing stock outs is to hold extra inventory (called Safety Stock).

In this case, we change the ROP formula to include that safety stock (ss).

ROP = d . Ld = 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 per year.

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

Probability of stock out is based on the following probability distribution:

Question: Find the Level of Safety Stock (ss) that minimizes the total additional holding cost and Stock out costs (annually).

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Example…

Stock out cost is an expected cost; That is:

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Example…

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

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

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

But there are stock out costs for two levels:1) If demand is 60 units at the ROP, Then a 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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Results of the example

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

Therefore, ROP = 50 + 20 = 70 units.

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Finding A Safety Stock Level

Managers may want to limit the possibility of stock out only to a small percentage, say 5%.

If demand level is assumed to be a normal distribution,

By using mean and standard deviation of 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 inventory item that has a normally distributed demand.

The mean demand is ( = 350) units and standard deviation is ( = 10).

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Example…

x : mean demand + safety stock x - z = ------

Using a standard Normal table we find z = 1.65 for 95% confidence.

We use the properties of a standardized normal curve to get a z value that corresponds to the.95 of the curve.

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Example…

Therefore,

x - ss

z = ------ = ---- = 1.65

ss = 16.5 units

Reorder point is elevated up by 16.5 units.

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Example…

8 Inventory Management

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