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Operations Management Inventory Management Chapter 12 - Part I

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Outline

- Functions of Inventory.
- ABC Analysis.
- Inventory Costs .
- Inventory Models for Independent Demand.
- Economic Order Quantity (EOQ) Model.
- Production Order Quantity (POQ) Model.
- Quantity Discount Model.
- Probabilistic Models with Varying Demand.
- Fixed Period Systems.

Types of Inventory

- Raw materials.
- Work-in-progress.
- Maintenance/repair/operating (MRO) supply.
- Finished goods.

The Functions of Inventory

- To ”decouple” or separate various parts of the production process.
- To smooth production (link supply and demand).
- To provide goods for customers (quick response).
- To take advantage of quantity discounts.
- Buy more to get a reduced price.
- To hedge against inflation and upward price changes (speculation).
- Buy more now if you think price will rise.

Disadvantages of Inventory

- High cost - $$$$$
- Money tied up in inventory could be better used elsewhere in the organization.
- Difficult to control.
- Inventories occur in many places.
- Hides production problems.
- Large inventories may overcome poor quality production or poor quality materials.

ABC Analysis

- Divide inventory into 3 classes based on annual $ volume.
- Annual $ volume = Annual demand x Unit cost.

A class - Most important.

15-20% of products. 60-80% of value.

B class -Less important.

20-40% of products. 15-30% of value.

C class - Least important.

50-60% of products. 5-10% of value.

ABC Analysis

- Sort products from largest to smallest annual $ volume.
- Divide into A, B and C classes.
- Focus on A products.
- Develop class A suppliers more.
- Give tighter physical control of A items.
- Forecast A items more carefully.
- Consider B products only after A products.

Classifying Items as ABC

25 products sorted by Annual $ Volume (Sales)

Product Sales %

1 100 14

2 92 13

3 88 12

4 60 8

5 58 8

6 53 7

7 49 7

8 41 6

9 32 4

10 26 4

11 21 3

12 18 2

13 16 2

14-25 66 9

Total 720

100

80

Annual $ Usage (x1000)

60

40

20

0

20

10

15

1

25

5

Product

Classifying Items as ABC

Class

% $ Vol

% Items

A

39%

12% (3/25)

100

B

52%

40% (10/25)

80

C

9%

48% (12/25)

Annual $ Usage (x1000)

60

A

40

20

B

C

0

80

40

60

0

100

20

% of Products

Inventory Accuracy

- Inventory accuracy importance:
- To determine when and how much to order.
- To achieve high level of service.
- Information system tracks inventory, but…
- Not all items sold are entered (scanned) properly.
- Some items disappear without being sold (theft, defective, damaged, etc.)

Inventory Counting

- Count products to verify inventory records.
- Shut down facility and count everything at one time (once per year).
- Cycle counting: count items continuously (count some each week).
- Count A items most frequently (for example, once a month).
- Count B items less frequently (twice a year).
- Count C items least frequently (once a year).

Inventory for Services

- Can be large $.
- “Shrinkage” (theft) is a problem.
- Often over 3%!
- Good personnel selection, training, and discipline is key.
- Establish tight control of shipments entering and leaving the facility.
- Enforce procedures for documenting product movement.
- Information systems can monitor inventory levels and help ensure accuracy.

Inventory Costs

- Holding costs - Associated with holding or “carrying” inventory over time.
- Ordering costs - Associated with costs of placing order and receiving goods.
- Setup costs - Cost to prepare a machine or process for manufacturing an order.
- Stockout costs- Cost of not making a sale and lost future sales.

Holding Costs

- Investment costs (borrowing, interest).
- Insurance.
- Taxes.
- Storage and handling.
- Extra staffing.
- Pilferage, damage, spoilage, scrap.
- Obsolescence.

Category

Investment costs

Housing costs

Material handling costs

Labor cost from extra handling

Pilferage, scrap, and obsolescence

Cost as a

% of Inventory Value

6 - 24%

3 - 10%

1 - 3.5%

3 -5%

2 - 5%

Inventory Holding Costs – Usually 20-30% of TotalSetup Costs

To change equipment and setup for new product:

- Clean-up costs.
- Re-tooling costs.
- Adjustment costs.
- etc.

Stockout Costs

For not making a sale and for lost future sales:

- Customer may wait for a backorder, or

- Cancel order (and acquire product elsewhere).

- Backorder costs: expediting, special orders, rush shipments, etc.
- Lost current sale cost.
- Lost future sales (hard to estimate).

Inventory Questions

- How much to order (each time)?
- 100 units, 50 units, 23.624 units, etc.
- When to order?
- Every 3 days, every week, every month, etc.
- When only 5 items are left, when only 10 items are left, when only 20 items are left, etc.
- Many different models can be used, depending on nature of products and demand.

Independent vs. Dependent Demand

- Independent demand - Demand for item is independent of demand for any other item.
- Dependent demand - Demand for item depends upon the demand for some other item.
- Example: Demand for car engines depends on demand for new cars.
- We will consider only independent demand.

Inventory Models

How much and when to order?

- Fixed order-quantity models.
- 1. Economic order quantity (EOQ).
- 2. Production order quantity (POQ).
- 3. Quantity discount.
- Probabilistic models.
- Fixed order-period models.

How Much and When to Order?

- Given a fixed annual demand for a product.
- With many small orders:
- Amount on hand is always small, so inventory is small.
- Frequent orders means cost of ordering is large.
- With few large orders:
- Amount on hand may be large (when order arrives), so inventory may be large.
- Infrequent orders mean cost of ordering is small.

EOQ – Economic Order Quanitity Models

- How much to order (each time)?
- Order size is a constant = Q
- Q is selected to minimize total cost.
- When to order?
- Order when amount remaining = ROP
- ROP is selected so chance of running out is small.

EOQ Assumptions

- Known and constant demand.
- Known and constant lead time.
- Instantaneous receipt of material.
- No quantity discounts.
- Only order cost and holding cost.
- No stockouts.

EOQ Model - How Much to Order?

Annual Cost

Total Cost Curve

Holding Cost Curve

Order Cost Curve

Order Quantity

Optimal Order Quantity (EOQ=Q*)

Why Holding Costs Increase

- For fixed annual demand, larger order quantities means:
- Larger inventory (larger amount ordered each time).
- Larger inventory holding cost.
- Example: Annual demand = 1200 units
- Order 600 each time.
- Maximum inventory = 600
- Order 50 each time.
- Maximum inventory = 50

Why Order Costs Decrease

- For fixed annual demand, larger order quantities means:
- Fewer orders per year.
- Smaller order cost per year.
- Example: Annual demand = 1200 units
- Order 600 each time.
- 1200/600 = 2 orders per year.
- Order 50 each time.
- 1200/50 = 24 orders per year.

Deriving an EOQ

- Develop an expression for total costs.
- Total cost = order cost + holding cost
- Find order quantity that gives minimum total cost (use calculus).
- Minimum is when slope is flat.
- Slope = Derivative.
- Set derivative of total cost equal to 0 and solve for best order quantity.

=

=

Expected Number of Orders per year

N

Q

D

S

Order Cost per year

=

Q

(average inventory level) H

=

Holding Cost per year

EOQ Model EquationsD = Annual demand (relatively constant)

S = Order cost per order

H = Holding (carrying) cost per unit per year

d = Demand rate (units per day, units per week, etc.)

L = Lead time (constant) (in days, weeks, hours, etc.)

Determine: Q = Order size (number of items per order)

Given

Order Quantity(Q)

AverageInventory (Q/2)

0

Time

EOQ Model - Average Inventory LevelMaximum inventory = Q Minimum inventory = 0

=

=

Expected Number of Orders per year

N

Q

D

S

Order Cost per year

=

Q

(average inventory level) H =

=

Holding Cost per year

EOQ Model EquationsD = Annual demand (relatively constant)

S = Order cost per order

H = Holding (carrying) cost per unit per year

d = Demand rate (units per day, units per week, etc.)

L = Lead time (constant) (in days, weeks, hours, etc.)

Determine: Q = Order size (number of items per order)

Given

Q

H

2

EOQ Model - How Much to Order?

Annual Cost

Total Cost Curve = (D/Q)S+(Q/2)H

Holding Cost =(Q/2)H

Order Cost Curve = (D/Q)S

Order Quantity

Optimal Order Quantity (EOQ=Q*)

1

S +

H = 0

Q2

2

D

Q

=

Total Cost

S +

H

Q

2

EOQ Total Cost OptimizationTake derivative of total cost with respect to Q and set equal to zero:

Solve for Q to get optimal order size:

×

×

2

D

S

EOQ = Q*

=

H

=

=

Expected Number of Orders

N

Q*

Working Days / Year

Expected Time Between Orders

=

=

T

N

EOQ Model EquationsD = Annual demand

S = Order cost per order

H = Holding (carrying) cost

×

×

2

D

S

Optimal Order Quantity

=

=

Q*

H

EOQ Model - When to order?

D = Annual demand (relatively constant)

d = Demand per day

L = Lead time in days

Determine: ROP = reorder point (number of pieces or items remaining when order is to be placed)

Given

D

Suppose demand is 10 per day and lead time is (always) 4 days.

When should you order?

When 40 are left!

=

d

Working Days / Year

=

×

ROP

d

L

Lead Time = time between placing and receiving an order

Reorder Point (ROP)

4th order

2nd order

3rd order

1st order received

1st order placed

EOQ Model - When To OrderInventory Level

Q*

Time

EOQ Example

Demand = 1200/year

Order cost = $50/order

Holding cost = $5 per year per item

260 working days per year

2 ×1200 ×50

=

= 154.92 units/order; so order 155 each time

Q*

5

1200/year

Expected Number of Orders = N =

= 7.74/year

155

260 days/year

Expected Time Between Orders = T =

= 33.6 days

7.74/year

1200

155

Total Cost =

50 +

5 = 387.10 + 387.50 = $774.60/year

155

2

Q

Total Cost =

50 +

5

Q

2

EOQ is RobustDemand = 1200/year

Order cost = $50/order

Holding cost = $5 per year per item

260 working days per year

Q = 155 units/order TC = $774.60/year

Q* = 154.92 units/order TC = $774.60/year = 387.30 + 387.30

Suppose we must order in multiples of 20:

Q = 140 units/order TC = $778.57/year (+0.5%)

Q = 160 units/order TC = $775.00/year (+0.05%)

Cost is nearly optimal!

Q

Total Cost =

50 +

5

Q

2

EOQ is RobustDemand = 1200/year

Order cost = $50/order

Holding cost = $5 per year per item

260 working days per year

Q = 155 units/order TC = $774.60/year

Q* = 154.92 units/order TC = $774.60/year = 387.30 + 387.30

Suppose we wish to order 6 times per year (every 2 months):

Q = 1200/6 = 200 units/order(200/order is 29% above Q*)

TC = $800.00/year = 300.00 + 500.00

(Cost is only 3.3% above optimal: $800 vs. $774.60)

EOQ Model is Robust

Annual Cost

Total Cost Curve

Small variation in cost

Order Quantity

154.92

Large variation in order size

Robustness

- EOQ amount can be adjusted to facilitate business practices.
- If order size is reasonably near optimal (+ or - 20%), then cost will be very near optimal (within a few percent).
- If parameters (order cost, holding cost, demand) are not known with certainty, then EOQ is still very useful.

EOQ Model - When to order?

Demand = 1200/year

Order cost = $50/order

Holding cost = $5 per year per item

260 working days per year

Lead time = 5 days

1200/year

=

= 4.615/day

d

260 days/year

ROP = 4.615 units/day 5 days = 23.07 units

-> Place an order whenever inventory falls to (or below) 23 units

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