12-1 Operations Management Inventory Management Chapter 12 - Part 2.

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Operations Operations ManagementManagement

Inventory ManagementInventory ManagementChapter 12 - Part 2Chapter 12 - Part 2

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OutlineOutline Functions of Inventory. ABC Analysis. Inventory Costs. Inventory Models for Independent Demand.

Economic Order Quantity (EOQ) Model. Production Order Quantity (POQ) Model. Quantity Discount Models.

Probabilistic Models for Varying Demand. Fixed Period Systems.

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Material is not received instantaneously. For example, it is produced in-house.

Other EOQ assumptions apply.

Model provides production lot size (like EOQ amount) for one product.

Similar to EOQ with setup cost rather than order cost.

Production Order Quantity ModelProduction Order Quantity Model

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Consider one product at a time.

Produce Q units in a production run; then switch and produce other products.

Later produce Q more units in 2nd production run (Q units of product of interest).

Later produce Q more units in 3rd production run, etc.

Production Order Quantity ModelProduction Order Quantity Model

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POQ Model Inventory LevelsPOQ Model Inventory Levels

Inventory LeveInventory Levell

TimeTimeProduction Production

BeginsBeginsProduction Production Run EndsRun Ends

Production portion of cycleProduction portion of cycle

Demand portion of cycle with no Demand portion of cycle with no production (of this product)production (of this product)

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POQ Model Inventory LevelsPOQ Model Inventory Levels

Inventory LeveInventory Levell

TimeTimeProduction Production

BeginsBeginsProduction Production Run EndsRun Ends

Production rate = p = 20/dayDemand rate = d = 7/day

Slope = -d = -7/day

Slope = p-d = 13/day

Note: Not all of production goes into inventory

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POQ Model Inventory LevelsPOQ Model Inventory Levels

Inventory LeveInventory Levell

TimeTimeProduction Production

BeginsBeginsProduction Production Run EndsRun Ends

Production rate = p = 20/dayDemand rate = d = 7/day

Slope = -d = -7/dayInventory decreases by 7/day after producing

Slope = p-d = 13/dayInventory increases by 13 each day while producing

Note: 1-(d/p) = fraction of production that goes into inventory

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Number of Production Runs per year = D Q

DD = Annual demand (relatively constant) = Annual demand (relatively constant)

SS = Setup cost per setup = Setup cost per setup

HH = Holding (carrying) cost per unit per year = Holding (carrying) cost per unit per year

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

pp = Production rate (units per day, units per week, etc.) = Production rate (units per day, units per week, etc.)

Determine: QDetermine: Q = Production run size (number of items per production run) = Production run size (number of items per production run)

POQ Model Equations POQ Model Equations

Setup Cost per year = SD

Q

Holding Cost per year = (average inventory level) H

Given

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POQ Model Inventory LevelsPOQ Model Inventory Levels

TimeTime

Inventory LevelInventory Level

Production Production Portion of CyclePortion of Cycle

Maximum Inventory Maximum Inventory = Q(1-(d/p))= Q(1-(d/p))

Demand portion of cycle with no supply

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Number of Production Runs per year = D Q

DD = Annual demand (relatively constant) = Annual demand (relatively constant)

SS = Setup cost per setup = Setup cost per setup

HH = Holding (carrying) cost per unit per year = Holding (carrying) cost per unit per year

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

pp = Production rate (units per day, units per week, etc.) = Production rate (units per day, units per week, etc.)

Determine: QDetermine: Q = Production run size (number of items per production run) = Production run size (number of items per production run)

POQ Model Equations POQ Model Equations

Setup Cost per year = SD

Q

Holding Cost per year = (ave. inventory level) H

Given

= H [1-(d/p)]Q 2

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Optimal Production Run Size =

Maximum inventory level = Q [1- (d/p)]

DD = Annual demand = Annual demand

SS = Setup cost per setup = Setup cost per setup

HH = Holding (carrying) cost per unit per year = Holding (carrying) cost per unit per year

dd = Demand rate = Demand rate

pp = Production rate = Production rate

POQ Model EquationsPOQ Model Equations

Total Cost = D Q

S + Q

2H [1-(d/p)]

=× ×

Q*D S

H[1-(d/p)]2 =

H2DS

p-dp

Given

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Run Length & Cycle LengthRun Length & Cycle Length

TimeTime

Inve

ntor

y Le

vel

Inve

ntor

y Le

vel

Production Run length (time) = Q /p

Cycle length (time) = Q /d

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POQ ExamplePOQ ExampleDemand = 1000/year (of product A)Setup cost = $100/setupHolding cost = $20 per year per itemProduction rate = 10/day365 working days per year

2 ×1000×100=Qp*20×[1-(2.74/10)]

= 117.36 units/run

Total Cost = 1000

117.36 100 +

117.36

220 [1-(2.74/10)]

Maximum inventory level = 117.36 [1- (2.74/10)] = 85.2 units

Demand rate = d = 1000/365 = 2.74/day

= 852.08 + 852.03 = $1704.11/year

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POQ ExamplePOQ ExampleDemand = 1000 units/yearProduction rate = 10 units/dayQp* = 117.36 units per run

Demand rate = d = 1000/365 = 2.74/day

Number of production runs per year = 1000/117.36 = 8.52

Cycle length = 117.36/(2.74/day) = 42.8 days

Production run length = 117.36/(10/day) = 11.74 days

11.74

42.8

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POQ is robust (like EOQ): Can adjust production run size. Useful even when parameters are uncertain. A large (20%) change in parameters or operations

will cause a small (~2%) change in total costs.

Production run size (Q) and run length (Q/p) can be adjusted to fit normal business cycles.

Robustness of POQRobustness of POQ

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Set production run length to 14 days (2 weeks) rather than 11.74 days (as was optimal).

Q/p = 14 days means that: Q = 10x14 = 140 units

Q = 140 is 19% over optimal value of 117.4 units.

Cycle length = Q/d = 140/2.74 = 51.1 days.

Total cost = $1730.68

Only 1.6% over minimum cost with optimal Q!

POQ Robustness ExamplePOQ Robustness Example

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POQ computes a production run size for a single product.

For multiple products made on the same equipment:1. Compute POQ, run time, and cycle time for each product.

2. Find a common cycle time for all products.

3. Recalculate run time and cycle time, so the common cycle time is a multiple of each product’s cycle time.

4. Fit production runs into largest cycle time.

POQ & Multiple ProductsPOQ & Multiple Products

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Example: Company makes 3 products: A, B, CA: Optimal run time = 3 days; Optimal cycle time = 10 days B: Optimal run time = 6 days; Optimal cycle time = 18 days C: Optimal run time = 10 days; Optimal cycle time = 33 days

Multiple Products ExampleMultiple Products Example

10 23

C

3 7 7 7 33

A AA

126 12

B6

B

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Optimal run time and cycle time:A: Run time = 3 days; Cycle time = 10 days (1 run/10 days)B: Run time = 6 days; Cycle time = 18 days (1 run/18 days)C: Run time = 10 days; Cycle time = 33 days (1 run/33 days)

Use 30 days as a common cycle; adjust run & cycle times:A: Run time = 3 days; Cycle time = 10 days (3 runs/30 days)B: Run time = 5 days; Cycle time = 15 days (2 runs/30 days)C: Run time = 9 days; Cycle time = 30 days (1 run/30 days)

Multiple Products ExampleMultiple Products Example

3 5 9 5 33

A B C B AA

2 days idle time

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Variation of EOQ (not POQ).

Allows quantity discounts. Reduced price for purchasing larger quantities.

Other EOQ assumptions apply.

Trade-off lower price to purchase item & increased holding cost from more items.

Total cost must include annual purchase cost.

Total Cost = Order cost + Holding cost + Purchase cost

Quantity Discount ModelQuantity Discount Model

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Holding cost: Depends on price.

Usually expressed as a % of price per unit time. 20% of price per year, 2% of price per month, etc.

I = Holding cost percent of price per year

P = Price per unit

H = Holding cost = IP

Quantity Discount Model - Holding Quantity Discount Model - Holding CostCost

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Order Quantity = =× ×

Q*D SIP

2

DD = Annual demand = Annual demand

SS = Order cost per order = Order cost per order

HH = Holding (carrying) cost = = Holding (carrying) cost = IPIP

I = I = Inventory holding cost % per yearInventory holding cost % per year

P = P = Price per unitPrice per unit

Quantity Discount EquationsQuantity Discount Equations

Total Cost ($/yr) = D Q

S + Q

2IP + PD

Annual purchase cost

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Quantity Discount ModelQuantity Discount Model

D = 1000/yearS = $100/orderI = 20% per year

Q P IP <500 $100 $20500-1000 $ 95 $19 1000 $ 90 $18

To solve:

1. Find EOQ amount for each discount level.

2. If EOQ is not in range for discount level, adjust to the nearest end of range.

3. Calculate total cost for each discount level.

4. Select lowest cost and corresponding Q.

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Quantity Discount ExampleQuantity Discount Example

D = 1000/yearS = $100/orderI = 20% per year

Q P IP <500 $100 $20500-1000 $ 95 $19 1000 $ 90 $18

1. P = $100 IP = $20

EOQ = 100 in range!

Total Cost = 1,000 + 1,000 + 100,000 = $102,000/year

2. P = $95 IP = $19

EOQ = 102.6 not in range (500-1000)!

Adjust to Q = 500

Total Cost = 200 + 4,750 + 95,000 = $99,950/year

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Quantity Discount Example - cont.Quantity Discount Example - cont.

D = 1000/yearS = $100/orderI = 20% per year

Q P IP <500 $100 $20500-1000 $ 95 $19 1000 $ 90 $18

3. P = $90 IP = $18

EOQ = 105.4 not in range (>1000)!

Adjust to Q = 1000

Total Cost = 100 + 9,000 + 90,000 = $99,100/year

Q Total costs<500 $102,100500-1000 $ 99,950 1000 $ 99,100 Lowest cost, so order 1000

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In basic EOQ model, demand and lead time are known and constant, so there should never be a stockout.

If demand or lead time vary, then may have a stockout: Due to larger than expected demand.

Due to longer than expected lead time.

StockoutsStockouts

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Probabilistic ModelsProbabilistic Models

Reorder Reorder Point Point (ROP)(ROP)

Time

Inventory Level

Lead TimeLead Time

Place Place orderorder

Receive Receive orderorder

Average demand

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Probabilistic Models - StockoutProbabilistic Models - Stockout

Reorder Reorder Point Point (ROP)(ROP)

Time

Inventory Level

Lead TimeLead Time

Place Place orderorder

Receive Receive orderorder

If demand is greater than average - then stockout

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Safety Stock to Reduce StockoutsSafety Stock to Reduce Stockouts

Time

Inventory Level

Lead TimeLead Time

Place Place orderorder

Receive Receive orderorder

Safety stock

New ROPNew ROP

Old ROP

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Safety stock is inventory held to protect against stockout.

Service level = 1 - Probability of stockout

Service level of 95% means 5% chance of stockout.

Higher service level means more safety stock.

More safety stock means higher ROP. ROP = Expected demand during lead time + Safety stock

Safety Stock & Service LevelSafety Stock & Service Level

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Demand follows normal distribution. d = Average demand rate per day.

= Standard deviation of demand.

ROP = d L + safety stock

safety stock = ss = Z

Z is from Standard Normal Table in Appendix I.

Probabilistic ModelsProbabilistic Models

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Order same amount every time = Q. Time between orders varies.

EOQ-based ModelsEOQ-based Models

ROPROP

TimeTimeTime between Time between 1st & 2nd order1st & 2nd order

Time between 2nd Time between 2nd & 3rd order& 3rd order

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Order at fixed intervals (e.g., every 2 weeks).

Order different amounts each time, based on amount on hand.

If large amount on hand, then order small amount.

If small amount on hand, then order large amount.

Useful when vendors visit routinely.

Example: P&G representative calls every 2 weeks.

Fixed Period ModelFixed Period Model

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Compute optimal order interval, T (equation is similar to EOQ). For example, 27.35 days

Compute maximum inventory level, M (equation is similar to ROP).

Adjust order interval to a convenient length. For example, one month.

Then, adjust M correspondingly.

Order M - inventory on hand every T time units.

Fixed Period ModelFixed Period Model

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Order at constant interval. Order amount Q varies: M - amount on hand.

Fixed Period ModelsFixed Period Models

On-hand On-hand for order 1for order 1

TimeTime1st 1st orderorder

2nd 2nd orderorder

3rd 3rd orderorder

On-hand On-hand for order 2for order 2