Inventory Models

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Inventory Models

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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


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• Order/Setup CostsThese 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


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:


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.


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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Inventory can be classified in various ways:

By Process By Importance By Shelf Life Raw materials Perishable Work in progress A, B, C Nonperishable Finished goods

By Process By Importance By Shelf Life Raw materials Perishable Work in progress A, B, C Nonperishable Finished goods

Used typically by accountants at manufacturing firms.Enables management to track the production process.

Items are classified by their relative importancein terms of the firm’s capital needs.

Management of items with short shelf life and long shelf life is very different

Inventory Classifications

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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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– Continuous review systems.• The system is continuously monitored.• A new order is placed when the inventory reaches a

critical point.

– EOQ

Review Systems continuous review system

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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:


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

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

juicers.


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

EOQ = sqrt((2*6240*12)/1,4) = 327,065

= 327

AAC – Solution:EOQ and Total Variable Cost

Savings of 16%

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

Savings of 16%

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TC(327) = 457.89 + 6240($10) + (13)($1.40) = $62,876.09

• Under the current ordering policy AAC holds 13 units safety stock (how come? observe):

• AAC is open 5 day a week.– The average daily demand = 120/week/5 = 24 juicers.– Lead time is 8 days. Lead time demand is (8)(24) = 192 juicers.– Reorder point without Safety stock = LD = 192.– Current policy: R = 205.– Safety stock = 205 – 192 = 13.

• For safety stock of 13 juicers the total cost is

TV(327) + Procurement + Safety stock cost holding cost

AAC – Solution:Reorder Point and Total Cost

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• Changing the order size

– Suppose juicers must be ordered in increments of 100 (order 300 or 400)– AAC will order Q = 300 juicers in each order.– There will be a total variable cost increase of $1.71.– This is less than 0.5% increase in variable costs.

• Changes in input parameters– Suppose there is a 20% increase in demand. D=7500 juicers.– The new optimal order quantity is Q* = 359.– The new variable total cost = TV(359) = $502 – If AAC still orders Q = 327, its total variable costs becomes

TV(327) = (327/2)($1.40) + (7500/327)($12) = $504.13

Only 0.4% increase

AAC – Solution:Sensitivity of the EOQ Results

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• For an order size of 327 juicers we have:– T = (327/ 6240) = 0.0524 year.

= 0.0524(52)(5) = 14 days.

– This is useful information because:

• Shelf life may be a problem.• Coordinating orders with other items might be

desirable.

AAC – Solution:Cycle Time

working days per week

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AAC – Excel Spreadsheet

=SQRT(2*$B$10*$B$14/$B$13)

=1/E11Copy to cell H12

=E10/B10Copy to cell H11

=$B$10*$B$11+E14+$B$13*B16Copy to Cell H15

=(E10/2)*$B$13+($B$10/E10)*$B$14Copy to cell H14

=$B$15*$B$10+$B$16-INT(($B$15*$B$10+$B$16)/E10)*E10

Copy to cell H13

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Service Levels and Safety Stocks

Service Levels and Safety Stocks

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Determining Safety Stock Levels

• Businesses incorporate safety stock requirements when determining reorder points.

• A possible approach to determining safety stock levels is by specifying desired service level .

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• The unit service level – The percentage of

demands that are filled without incurring any delay.

– Applied when the percentage of unsatisfied demand should be under control.

Two Types of Service Level

• The cycle service level – The probability of not

incurring a stockout during an inventory cycle.

– Applied when the likelihood of a stockout, and not its magnitude, is important for the firm.

Service levels can be viewed in two ways.

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• In many cases short run demand is variable even though long run demand is assumed constant.

• Therefore, stockout events during lead time may occur unexpectedly in each cycle.

• Stockouts occur only if demand during lead time is greater than the reorder point.

The Cycle Service Level Approach

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• To determine the reorder point we need to know:– The lead time demand distribution.– The required service level.

• In many cases lead time demand is approximately normally distributed. For the normal distribution case the reorder point is calculated by


R = mL + zasL 1 – a = service level

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m=192

P(DL> R) = P(Z > (R – mL)/sL) = a. SinceP(Z > Za) = a, we have Za = (R – mL)/sL,

which gives…


P(DL>R) = a Service level = P(DL<R) = 1 – a

R

R = mL + zasL

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• Assume that lead time demand is normally distributed.

• Estimation of the normal distribution parameters:

– Estimation of the mean weekly demand = ten weeks average demand = 120 juicers per week.

– Estimation of the variance of the weekly demand = Sample variance = 83.33 juicers2.

AAC - Cycle Service Level Approach

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• To find mLand sL the parameters m (per week) and s (per week) must be adjusted since the lead time is longer than one week.

– Lead time is 8 days =(8/5) weeks = 1.6 weeks.

• Estimates for the lead time mean demand and variance of demand

mL » (1.6)(120) = 192; s2L » (1.6)(83.33) =

133.33

AAC - Cycle Service Level Approach

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• Let us use the current reorder point of 205 juicers.

205 = 192 + z (11.55) z = 1.13

• From the normal distribution table we have that a

reorder

point of 205 juicers results in an 87% cycle service level.

133 33.

AAC - Service Level for a given Reorder Point

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• Management wants to improve the cycle service level to 99%.

• The z value corresponding to 1% right hand tail is 2.33.

R = 192 + 2.33(11.55) = 219 juicers.

• Management wants to improve the cycle service level to 99%.

• The z value corresponding to 1% right hand tail is 2.33.

R = 192 + 2.33(11.55) = 219 juicers.

AAC – Reorder Point for a given Service Level

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• AAC is willing to run out of stock an average of at most one cycle per year with an order quantity of 327 juicers.

• What is the equivalent service level for this strategy?

• AAC is willing to run out of stock an average of at most one cycle per year with an order quantity of 327 juicers.

• What is the equivalent service level for this strategy?

AAC – Acceptable Number of Stockouts per Year

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AAC – Acceptable Number of Stockouts per Year

• There will be an average of

6240/327 = 19.08 lead times per year.

• The likelihood of stockouts = 1/19 = 0.0524.• This translates into a service level of 94.76%

• There will be an average of

6240/327 = 19.08 lead times per year.

• The likelihood of stockouts = 1/19 = 0.0524.• This translates into a service level of 94.76%

• When lead time demand follows a normal distributionservice level can be calculated as follows:– Determine the value of z that satisfy the equation

L(z) = aQ* / sL

– Solve for R using the equation

R = mL + zsL

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The Unit Service Level Approach

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=NORMDIST(B8,B5,B6,TRUE)

AAC – Cycle Service Level (Excel spreadsheet)

=NORMINV(B7,B5,B6)

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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.

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.

EOQ Models with Quantity Discounts

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

TC(Q) = (Q/2)Ch + (D/Q)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 Schedule

1-299 $10.00300-599 $9.75600-999 $9.40

1000-4999 $9.505000 $9.00

Quantity Discount Schedule

1-299 $10.00300-599 $9.75600-999 $9.40

1000-4999 $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



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

999600

Q3*Q3*

336$9.50

AAC – All Units Discount Procedure

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– Step 2 : Modify Q i *



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



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*

999600

Q3*Q3*

336$9.50


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



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



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


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Calculation of Optimal Inventory Policy Under All-Units Quantity Discounts

OPTIMALINPUTS Values OUTPUTS Values

Annual Demand, D = 6240.00 Order quantity, Q* = 5000Per Unit Cost, C = 10.00 Cycle Time (in years), T = 0.801282051Annual Holding Cost Rate, H = 0.14 # of Cycles Per Year, N = 1.248Annual Holding Cost Per Unit, Ch = 1.40 Reorder Point, R = 205.0000Order Cost, Co = 12.00 Total Annual Cost, TC(Q*) = 59341.36Lead Time (in years), L = 0.03077 Safety Stock, SS = 13.00

DISCOUNTSLevel Breakpoint Discount Price Q* TC(Q*) Modified Q*

0 1 10.00 327 62876.09 3271 300 9.75 331 61309.88 3312 600 9.50 336 59821.09 6003 1000 9.40 337 59405.99 10004 5000 9.00 345 59341.36 50005678

AAC – All Units Discount Excel Worksheet

Inventory Models

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