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Inventory ModelsInventory Models

Chapter 8

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8.1 Overview of Inventory Issues

• Proper control of inventory is crucial to the success of an enterprise.

• Typical inventory problems include:– Basic inventory – Planned shortage – Quantity discount – Periodic review– Production lot size – Single period

• Inventory models are often used to develop an optimal inventory policy, consisting of:– An order quantity, denoted Q.– A reorder point, denoted R.

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• Inventory analyses can be thought of as cost-control techniques.

• Categories of costs in inventory models:– Holding (carrying costs)– Order/ Setup costs– Customer satisfaction costs– Procurement/Manufacturing costs

Type of Costs in Inventory Models

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• Holding Costs (Carrying costs): These costs depend on the order size– Cost of capital – Storage space rental cost– Costs of utilities– Labor– Insurance– Security– Theft and breakage– Deterioration or Obsolescence

Ch = Annual holding cost per unit in inventoryH = Annual holding cost rateC = Unit cost of an item

Ch = H * C

Type of Costs in Inventory Models

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• Order/Setup Costs

These costs are independent of the order size.– Order costs are incurred when purchasing a good

from a supplier. They include costs such as• Telephone • Order checking• Labor • Transportation

– Setup costs are incurred when producing goods for sale to others. They can include costs of

• Cleaning machines• Calibrating equipment• Training staff

Type of Costs in Inventory Models

Co = Order cost or setup cost

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• Customer Satisfaction Costs– Measure the degree to which a

customer is satisfied.– Unsatisfied customers may:

• Switch to the competition (lost sales).• Wait until an order is supplied.

– When customers are willing to wait there are two types of costs incurred:

Type of Costs in Inventory Models

Cb= Fixed administrative costs of an out of stock item ($/stockout unit).

Cs = Annualized cost of a customer awaiting an out of stock item($/stockout unit per year).

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• Procurement/Manufacturing Cost– Represents the unit purchase cost

(including transportation) in case of a purchase.

– Unit production cost in case of in-house manufacturing.

Type of Costs in Inventory Models

C = Unit purchase or manufacturing cost.

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• Demand is a key component affecting an inventory policy.

• Projected demand patterns determine how an inventory problem is modeled.

• Typical demand patterns are:– Constant over time (deterministic inventory models)– Changing but known over time (dynamic models)– Variable (randomly) over time (probabilistic models)

Demand in Inventory Models

D = Demand rate (usually per year)

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• Two types of review systems are used:– Continuous review systems.

• The system is continuously monitored.• A new order is placed when the inventory reaches a critical

point.– Periodic review systems.

• The inventory position is investigated on a regular basis.• An order is placed only at these times.

Review Systems

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• The item has a sufficiently long shelf life.• The item is monitored using a continuous review

system.• All the cost parameters remain constant forever

(over an infinite time horizon).• A complete order is received in one batch.

8.2 Economic Order Quantity Model - Assumptions

• Demand occurs at a known and reasonably constant rate.

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• The constant environment described by the EOQ assumptions leads to the following observation:

The optimal EOQ policy consists of same-size orders.

Q QQ

The EOQ Model – Inventory profile

This observation results in the following inventory profile :

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Q QQ

Total Annual Inventory Costs

= Total Annual Holding Costs

Total Annual ordering Costs

Total Annual procurement Costs

++

TC(Q) = (Q/2)Ch + (D/Q)Co + DC

ChCh

The optimal order SizeThe optimal order Size

2DCo2DCoQ* = Q* =

Cost Equation for the EOQ Model

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Constructing the total annual variable cost curve

Total Holding Costs

Total ordering costs

Add the two curves to one anotherTotal annual holding and ordering costs

Q

TV(Q)

Q*

The optimal order size

o* * * * *

TV(Q) and Q*

Note: at the optim

al order size

total holding costs and ordering costs

are equal

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The curve is reasonably flat around Q*.

Q*

Deviations from the optimal order size cause only small increase in the total cost.

Sensitivity Analysis in EOQ models

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• The cycle time, T, represents the time that

elapses between the placement of orders.

• Note, if the cycle time is greater than the shelf life, items will go bad, and the model must be modified.

T = Q/D

Cycle Time

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• To find the number of orders per years take the reciprocal of the cycle time

N = D/Q

• Example: The demand for a product is 1000 units per year. The order size is 250 units under an EOQ policy.• How many orders are placed per year? N = 1000/250 = 4 orders.• How often orders need to be placed (what is the cycle time)?

T = 250/1000 = ¼ years. {Note: the four orders are equally spaced}.

Number of Orders per Year

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• In reality lead time always exists, and must be accounted for when deciding when to place an order.

• The reorder point, R, is the inventory position when an order is placed.

• R is calculated by

L and D must be expressed in the same time unit.

R = L DR = L D

Lead Time and the Reorder Point

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Inventory position

LPlace the order now

Reorder Point

R = Inventory at hand at the beginning of lead time

Lead Time and the Reorder Point –Graphical demonstration: Short Lead Time

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Outstanding order

Place the order now

R = inventory at hand at the beginning of lead time + one outstanding order = demand during lead time = LD

Inventory at

hand

L

Lead Time and the Reorder Point –Graphical demonstration: Long Lead Time

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• Safety stocks act as buffers to handle:– Higher than average lead time demand.– Longer than expected lead time.

• With the inclusion of safety stock (SS), R is calculated by

• The size of the safety stock is based on having a desired service level.

R = LD + SS

Safety stock

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LPlace the order now

Reorder Point

R = LD

Safety stockPlanned situation

Actualsituation

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L

R = LD

Safety stock

Actualsituation

+ SS

Reorder Point

Place the order now

SS=Safety stock

?

The safety stockprevents excessiveshortages.

LD

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Inventory Costs Including safety stock

Total Annual Inventory Costs

= Total Annual Holding Costs

Total Annual ordering Costs

Total Annual procurement Costs

++

TC(Q) = (Q/2)Ch + (D/Q)Co + DC + ChSS

Safety stockholding cost

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ALLEN APPLIANCE COMPANY (AAC)

• AAC wholesales small appliances.

• AAC currently orders 600 units of the Citron brand juicer each time inventory drops to 205 units.

• Management wishes to determine an optimal ordering policy for the Citron brand juicer

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Sales of Juicers over the last 10 weeksWeek 1 2 3 4 5Sales 105 115 125 120 125Week 6 7 8 9 10Sales 120 135 115 110 130

• Data– Co = $12 ($8 for placing an order) + (20 min. to check)($12 per hr) – Ch = $1.40 [HC = (14%)($10)]– C = $10.– H = 14% (10% ann. interest rate) + (4% miscellaneous)– D = demand information of the last 10 weeks was collected:

ALLEN APPLIANCE COMPANY (AAC)

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• Data– The constant demand rate seems to be a good

assumption.– Annual demand = (120/week)(52weeks) = 6240 juicers.

ALLEN APPLIANCE COMPANY (AAC)

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• Current ordering policy calls for Q = 600 juicers.TV( 600) = (600 / 2)($1.40) + (6240 / 600)($12) = $544.80

• The EOQ policy calls for orders of size

AAC – Solution:EOQ and Total Variable Cost

Savings of 16%

2(6240)(12)1.40 = 327.065 327 =Q*

TV(327) = (327 / 2)($1.40) + (6240 / 327) ( $12) = $457.89

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• Quantity Discounts are Common Practice in Business– By offering discounts buyers are encouraged to increase

their order sizes, thus reducing the seller’s holding costs.

– Quantity discounts reflect the savings inherent in large orders.

– With quantity discounts sellers can reward their biggest customers without violating the Robinson - Patman Act.

8.4 EOQ Models with Quantity Discounts

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• Quantity Discount Schedule– This is a list of per unit discounts and their corresponding

purchase volumes.– Normally, the price per unit declines as the order quantity

increases.– The order quantity at which the unit price changes is called a

break point.– There are two main discount plans:

• All unit schedules - the price paid for all the units purchased is based on the total purchase.

• Incremental schedules - The price discount is based only on the additional units ordered beyond each break point.

8.4 EOQ Models with Quantity Discounts

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• To determine the optimal order quantity, the total purchase cost must be included

TC(Q) = (Q2)Ch + (DQ)Co + DCi + ChSS

Ci represents the unit cost at the ith pricing level.

All Units Discount Schedule

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AAC - All Units Quantity Discounts

Quantity Discount Schedule1-299 $10.00300-599 $9.75600-999 $9.401000-4999 $9.505000 $9.00

Quantity Discount Schedule1-299 $10.00300-599 $9.75600-999 $9.401000-4999 $9.505000 $9.00

• AAC is offering all units quantity discounts to its customers.

• Data

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Should AAC increase its regular order of 327 juicers, to take advantage of the discount?Should AAC increase its regular order of 327 juicers, to take advantage of the discount?

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AAC – All units discount procedure

– Step 1: Find the optimal order Qi* for each discount level

“i”. Use the formula– Step 2: For each discount level “i” modify Q i

* as follows• If Qi

* is lower than the smallest quantity that qualifies for the i th discount, increase Qi

* to that level.

• If Qi* is greater than the largest quantity that qualifies for the ith discount,

eliminate this level from further consideration.

– Step 3: Substitute the modified Q*i value in the total cost formula

TC(Q*i ).

– Step 4: Select the Q i

* that minimizes TC(Q i*)

Q DC Co h* ( ) / 2

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Step 1: Find the optimal order quantity Qi* for each

discount level “i” based on the EOQ formula

Lowest cost order size per discount levelDiscount Qualifying Price

level order per unit Q*0 1-299 10.00 3271 300-599 9.75 3312 600-999 9.50 3363 1000-4999 9.40 3374 5000 9.00 345

AAC – All units discount procedure

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– Step 2 : Modify Q i *

Modified Q* and total CostQualified Price Modified Total

Urder per Unit Q* Q* Cost1-299 10.00 300 **** ****

300-599 9.75 331 331 61,292.13600-999 9.50 336 600 59,803.80

1000-4999 9.40 337 1000 59,388.885000 9.00 345 5000 59,324.98

Modified Q* and total CostQualified Price Modified Total

Urder per Unit Q* Q* Cost1-299 10.00 300 **** ****

300-599 9.75 331 331 61,292.13600-999 9.50 336 600 59,803.80

1000-4999 9.40 337 1000 59,388.885000 9.00 345 5000 59,324.98

1 299

Q1*Q1*

300

$10/unit

599331

Q2*Q2*

$9.75/unit

999999600

Q3*Q3*

336

$9.50

AAC – All Units Discount Procedure

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– Step 2 : Modify Q i *

Modified Q* and total CostQualified Price Modified Total

Urder per Unit Q* Q* Cost1-299 10.00 300 **** ****

300-599 9.75 331 331 61,292.13600-999 9.50 336 600 59,803.80

1000-4999 9.40 337 1000 59,388.885000 9.00 345 5000 59,324.98

Modified Q* and total CostQualified Price Modified Total

Urder per Unit Q* Q* Cost1-299 10.00 300 **** ****

300-599 9.75 331 331 61,292.13600-999 9.50 336 600 59,803.80

1000-4999 9.40 337 1000 59,388.885000 9.00 345 5000 59,324.98

1 299

Q1*Q1*

300

$10/unit

331

Q2*Q2*

999999600

Q3*Q3*

336

$9.50

AAC – All Units Discount Procedure

Q3*Q3* Q3

*Q3* Q3

*Q3* Q3

*Q3* Q3

*Q3*Q3

*Q3*

Q3*Q3*

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– Step 3: Substitute Q I * in the total cost function

– Step 4

Modified Q* and total CostQualified Price Modified Total

Urder per Unit Q* Q* Cost1-299 10.00 300 **** ****

300-599 9.75 331 331 61,292.13600-999 9.50 336 600 59,803.80

1000-4999 9.40 337 1000 59,388.885000 9.00 345 5000 59,324.98

Modified Q* and total CostQualified Price Modified Total

Urder per Unit Q* Q* Cost1-299 10.00 300 **** ****

300-599 9.75 331 331 61,292.13600-999 9.50 336 600 59,803.80

1000-4999 9.40 337 1000 59,388.885000 9.00 345 5000 59,324.98

AAC should order 5000 juicersAAC should order 5000 juicers

AAC – All Units Discount Procedure