1

Inventory Control

2

Week 1 Introduction to Production Planning and Inventory Control

Week 2 Inventory Control – Deterministic Demand Week 3 Inventory Control – Stochastic Demand Week 4 Inventory Control – Stochastic Demand Week 5 Inventory Control – Stochastic Demand Week 6 Inventory Control – Time Varying Demand Week 7 Inventory Control – Multiple Echelons

Lecture Topics

3

Week 8 Production Planning and Scheduling Week 9 Production Planning and Scheduling Week 12 Managing Manufacturing Operations Week 13 Managing Manufacturing Operations Week 14 Managing Manufacturing Operations Week 10 Demand Forecasting Week 11 Demand Forecasting Week 15 Project Presentations

Lecture Topics (Continued…)

4

Inventory in the US Economy

$290.40 billionWholesale(23.2%)

$122.10billionOther(9.8%

$98.60 billionFarm

(7.9%)

$424.60 billionManufacturing

(33.9%)

$316.00 billionRetail (25.2%)

5

Inventory Types

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Inventory Types

Raw Materials

Work-in-process (WIP)

Finished goods inventory (FGI)

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Inventory Location

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Inventory Location

Manufacturing Facility

In-Transit

Warehouse

Retailer

Customer

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Benefits of Inventory

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Benefits of Inventory

Reduces ordering, setup & transportation costs (economies of scale)

Buffer against demand fluctuations Buffer against supply fluctuations

Supply shortages Variability in supply lead times

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Benefits of Inventory (continued…)

Buffer against price fluctuations Benefits from quantity discounts Protects production capacity Allows production smoothing Reduces managerial complexity (eliminates

the need for coordination) Can increase demand

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The Cost of Inventory

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The Cost of Inventory Tied up capital Warehousing cost Deterioration Obsolescence Demand shortfall Quality defects Changes in raw material prices Changes in product design specifications

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What if demand uncertainty and variability are eliminated?

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What if demand uncertainty and variability are eliminated? What if replenishment lead times are made insignificant?

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What if demand uncertainty and variability are eliminated? What if replenishment lead times are made insignificant? What if ordering & setup costs are made negligible?

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What if demand uncertainty and variability are eliminated? What if replenishment lead times are made insignificant? What if ordering & setup costs are made negligible? What if production capacity is never a constraint?

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Demand Constant Time varying Stochastic

Supply lead time Deterministic Stochastic Load-dependent

Review Continuous review Periodic review

Characteristics of Inventory Systems

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Excess Demand Backordering Lost sales Impatient customers Item substitution

Capacity Unlimited Limited Deterministic Stochastic

Characteristics of Inventory Systems(continued…)

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Number of items & customer classes Single item Multiple items Single customer class Multiple customer classes

Inventory quality Perishability Obsolescence Imperfect yield

Characteristics of Inventory Systems(continued…)

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Holding cost (h) Capital cost Taxes and insurance Deterioration, spoilage, obsolescence

Ordering (setup) cost (A) Purchasing (production) costs (c) Shortage cost (p)

backordering cost lost sale cost

Cost Measures

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h = ic (cost per unit per time period)28% = cost of capital 2% = taxes and insurance6% = storage cost1% = breakage cost 37% = total interest charge (i) per year

If c = $100, then h = ic = $37

Example

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The Economic Order Quantity (EOQ) Model

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Assumptions of the Basic Model

Demand occurs continuously over time with a constant rate Inventory can be replenished instantaneously There are no capacity limits or limits on the size of

replenishment orders A replenishment order incurs a fixed ordering (or setup)

cost Multiple products can be analyzed independently of each

other No backorders are allowed

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Notation

D: demand rate (units per unit time)

c: unit purchase/production cost (dollars per unit)

A: fixed cost to place an order (dollars)

h: holding cost (dollars per unit per unit time); if the holding cost consists entirely of interest on money tied up in inventory, then h=ic where i is an interest rate.

Q: the size of the order (a decision variable)

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Inventory versus Time

Q

Inve

ntor

y

Time

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Inventory versus Time

Q/D 2Q/D 3Q/D 4Q/D

Q

Inve

ntor

y

Time

28

Costs Holding cost:

average inventory 2Average holding cost per unit time 2

Average unit holding cost 2

Q

hQ

hQD

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Costs Holding cost:

Ordering/setup cost: per order, so unit order cost

Average order cost per unit time=

AA QADQ

average inventory 2Average holding cost per unit time 2

Average unit holding cost 2

Q

hQ

hQD

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Costs Holding cost:

Ordering/setup cost:

Production cost: c per unit

per order, so unit order cost

Average order cost per unit time=

AA QADQ

average inventory 2Average holding cost per unit time 2

Average unit holding cost 2

Q

hQ

hQD

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Total Cost

Total cost per unit time:

cDQDAhQQY

2)(

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Total Cost

Total cost per unit time:

Total cost per unit:'( )

2hQ AY Q cD Q

cDQDAhQQY

2)(

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Economic Order Quantity

hQ/2

AD/Q

Q

Y(Q

)

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The Economic Order Quantity

2

( ) 02

dY Q h AdQ D Q

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The Economic Order Quantity

hADQ

QA

Dh

dQQdY

2

02

)(

*

2

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Optimal Cost

Optimal average cost per unit:

Optimal average cost per unit time (e.g., per year):

**

*'( )

2

22 22 2 /

2

hQ AY Q cD Q

h AD h A cD AD hA c Ah D cAD h

*( ) 2Y Q ADh cD

37

Sensitivity to Order Quantity

Ordering and holding cost from using Q’:

Ratio

2hQ AD

Q

*

* *

Cost( ) 2 1Cost( ) 22

Q hQ AD Q Q QQ Q QADh

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Sensitivity to Order Quantity

Ordering and holding cost from using Q’:

Ratio

Example Q' = 2Q*, then the ratio of the actual to optimal cost is(1/2)[2 + (1/2)] = 1.25

2hQ AD

Q

*

* *

Cost( ) 2 1Cost( ) 22

Q hQ AD Q Q QQ Q QADh

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Sensitivity to Order Quantity

Large deviations from the optimal order quantity lead to relatively small deviations from the optimal cost.

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Order Quantity versus Order Interval

Order Interval: Let T represent time between orders, then

Total cost:

Optimal Order Interval:

DQT

** 2A QT

hD D

cDTAhDTcD

QDAhQQY

22)(

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Some Limitations of the EOQ Model

Demand is deterministic and constant Instantaneous replenishments Ordering costs are constant and independent of

order size No accounting for interactions among multiple items No backordering

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Extensions

Non-zero order replenishment lead times

Non-zero safety stocks

Finite supply capacity

Backordering

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Non-Zero Replenishment Lead Times

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Non-Zero Replenishment Lead Times

If L is the lead time and r is the reorder point, then r = DL

A non-zero lead time has no effect on Q* or Y(Q)

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Non-Zero Safety Stocks

If ss is the safety stock, then

A non-zero safety stock affects Y(Q) but has no effect on Q*

cDQDAssQhQY )2()(