Inventory Management MD707 Operations Management Professor Joy Field.

Inventory Management

MD707 Operations Management

Professor Joy Field

Types of Inventory

Cycle inventory

Safety stock

Anticipation inventory

Pipeline inventory (WIP, finished goods, goods-in-transit)

Replacement parts, tools, and supplies

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Functions of Inventory

To meet anticipated customer demand

To smooth seasonal requirements

To decouple operations

To protect against stockouts

To take advantage of order cycles

To hedge against price increases

As a result of operations cycle and throughput times

To take advantage of quantity discounts

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Managing Independent Demand Inventory

Managing independent demand inventory involves answering two questions:

How much to order? When to order?

When answering these questions, a manager needs to consider costs (holding or carrying costs, ordering costs, shortage costs, unit costs) and the tradeoff between costs and customer service. Inventory management efforts may be allocated based on the relative importance of an item, determined through a classification system such as an A-B-C approach.

Customer satisfaction and inventory turnover (i.e., the ratio of annual cost of goods sold to average inventory investment) are two measures of inventory management effectiveness.

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EOQ Assumptions

Item independence

Demand is known and constant

Lead time does not vary

Each order is received in a single delivery

There are no quantity discounts

Only two relevant costs (holding and ordering costs)

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EOQ Inventory CycleDemand is Known and Constant

Profile of Inventory Level Over Time

Quantityon hand

Q

Receive order

Placeorder

Receive order

Placeorder

Receive order

Lead time

Reorderpoint

Usagerate

Time

12-6

Total Annual Cost

Annual carrying cost Annual carrying cost =

Annual ordering cost Annual ordering cost =

Total annual cost:

)(H2

Q

)(SQ

D

)()( SQD

H2Q

TC

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Derivation of Economic Order Quantity (EOQ)

and Time Between Orders (TBO)

Total annual cost: TC =

Take the first derivative of cost with respect to

quantity:

Setting and solving for Q:

Time between orders:

)()(2

SQ

DH

Q

)(2 2

SQ

DH

dQ

dTC

0dQ

dTCH

DSQEOQ o

2,

DQ

TBOQ

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EOQ Cost Curves

Order Quantity (Q)

The Total-Cost Curve is U-Shaped

Ordering Costs

Q*

An

nu

al C

ost

(optimal order quantity)

Holding Costs

SQ

DH

QTC

2

12-9

Overland Motors Example

What is the annual cost of the current policy of using a 1,000-unit lot size?

What is the order quantity that minimizes cost?

What is the time between orders for the quantity in part b?

If the lead time is two weeks, what is the reorder point?

Overland Motors uses 25,000 gear assemblies each year (i.e. 52 weeks) and purchases them at $3.40 per unit. It costs $50 to process and receive each order, and it costs $1.10 to hold one unit in inventory for a whole year. Assume demand is constant. The purchasing agent has been ordering 1,000 gear assemblies at a time, but can adjust his order quantity if it will lower costs.

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Economic Production Quantity (EPQ)

Maximum Cycle Inventory

Total cost = Annual holding cost + Annual ordering cost

Economic Production Quantity (EPQ)

Similar to the EOQ but used for batch production. A complete order is no longer received at once and inventory is replenished gradually (i.e., non-instantaneous replenishment).

)()(max p

upQup

p

QI

)())((2

)()(2

max SQD

Hp

upQS

QD

HI

TC

upp

HDS

Qo 2

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EPQ Inventory CycleNoninstantaneous Replenishment

Q

Q*

Imax

Productionand usage

Productionand usage

Productionand usage

Usageonly

Usageonly

Cumulativeproduction

Amounton hand

Time

12-12

EPQ Example

What is the economic production quantity?

How long is the production run?

What is the average quantity in inventory?

What are the total annual costs associated with the EPQ?

A domestic automobile manufacturer schedules 12 two-person teams to assemble 4.6 liter DOHC V-8 engines per work day. Each team can assemble five engines per day. The automobile final assembly line creates an annual demand for the DOHC engine at 10,080 units per year. The engine and automobile assembly plants operate six days per week, 48 weeks per year. The engine assembly line also produces SOHC V-8 engines. The cost to switch the production line from one type of engine to the other is $100,000. It costs $2,000 to store one DOHC V-8 for one year.

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Quantity Discounts

In the case of quantity discounts (price incentives to purchase large quantities), the unit price, P, is relevant to the calculation of total annual cost (since the price is no longer fixed).

Total cost = Annual holding cost + Annual ordering cost + Annual cost of materials

PDSQD

HQ

TC )()(2

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Quantity DiscountsTwo-Step Procedure

Step 1 Beginning with lowest price, calculate the EOQ for each price

level until a feasible EOQ is found. It is feasible if it lies in the range corresponding to its price.

Step 2 If the first feasible EOQ found is for the lowest price level, this

quantity is best. Otherwise, calculate the total cost for the first feasible EOQ and for the larger price break quantity at each lower price level. The quantity with the lowest total cost is optimal.

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Total Cost Curves with Quantity Discounts

12-16

Quantity Discounts Example

Step 1 =

=

Step 2 =

=

Order Quantity Price Per Unit 1-99 $50

100 or more $45

If the ordering cost is $16 per order, annual holding cost is 20% of the per unit purchase price, and annual demand is 1,800 items, what is the best order quantity?

00.45EOQ

00.50EOQ

___TC

___TC

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Perpetual (Continual) Inventory Review System

A perpetual (continual) inventory review system tracks the remaining inventory of an item each time a withdrawal is made, to determine if it is time to reorder.

Decision rule: Whenever a withdrawal brings the inventory down to the reorder point (ROP), place an

order for Q (fixed) units.

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Variations of the Perpetual Inventory SystemBased on the characteristics of the lead time demand and the lead time, the perpetual inventory

system is implemented by ordering the EOQ, , at the ROP as follows:

Lead time demand (dLT) Lead time (LT) Approach Known and constant Known and constant Order oQ when ROP is equal to the lead time demand.

Variable, normally distributed, average lead time demand and dLT known

Variable, normally distributed, dLT known

Calculate the safety stock for a given service level (= z dLT ) using the table on p.576 or pp.882-3 to determine z.*

Unknown, but average daily or weekly demand and d known, normally distributed

Known and constant Calculate the expected lead time demand by multiplying the average daily or weekly demand by the lead time. Calculate the safety stock for a given service level (= z dLT ) with z determined as above.*

Unknown, but daily or weekly demand known and constant

Variable, normally distributed, average lead time and LT known

Calculate the expected lead time demand by multiplying the daily or weekly demand by the average lead time. Calculate the safety stock for a given service level (= zd LT ) with z determined as above.*

Unknown, but average daily or weekly demand and d known, normally distributed

Variable, normally distributed, average lead time and LT known

Calculate the expected lead time demand by multiplying the average daily or weekly demand by the average lead time. Calculate the safety stock for a given service level

(= 222LTd dLTz ) with z determined as above.*

*Order oQ when ROP is equal to the expected lead time demand plus safety stock.

oQ

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Reorder Point

ROP

Risk ofstockoutService level

Expecteddemand

Safetystock

0 z

Quantity

z-scale

The ROP based on a normaldistribution of lead time demand

12-20

Shortages and Service Levels

dLTzEnE )()(

The ROP calculation relates the probability of being able to satisfy demand during the lead time for ordering. In order to determine the expected amount of units to be short during this period, calculate:

where: E(n) = Expected number of units short per order cycle,E(z) = Standardized number of units short (obtained from Table 12.3, p.576), = Standard deviation of lead time demand To calculate the expected number of units short per year:

where: E(N) = Expected number of units short per year, D = Yearly demand, Q = Order size

dLT

Q

DnENE )()(

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Perpetual Inventory System Example

What is the EOQ for this item? What is the desired safety stock? What is the desired reorder point? What is the expected number of units short each cycle and per year?

You are reviewing the company’s current inventory policies for its perpetual inventory system, and began by checking out the current policies for a sample of items. The characteristics of one item are:

Average demand = 10 units/wk (assume 52 weeks per year)Ordering cost (S) = $45/orderHolding cost (H) = $12/unit/yearMean lead time demand = 30 unitsStandard deviation of lead time demand = 17 unitsService-level = 70%

If instead of the above situation, suppose the lead time is known and constant at 2 weeks and the standard deviation of lead time demand is unknown. However, we do know the standard deviation of weekly demand to be 10 units. How do your answers change?

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The Single-Period Model

Calculate the shortage and excess costs Cshortage = Cs = Revenue per unit – Cost per unit

Cexcess = Ce = Original cost per unit – Salvage value per unit Calculate the service level, which is the probability that

demand will not exceed the stocking level

Service level =

Determine the optimal stocking level, , using the service level and demand distribution information

Used to handle ordering of perishables and items that have a limited useful life

Analysis of single-period situations generally focuses on two costs: shortage costs (i.e., unrealized profit per unit) and excess costs (the cost per unit less any salvage cost)

es

s

CC

C

do zdS

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Single-Period Problem

The concession manager for the college football stadium must decide how many hot dogs to order for the next game. Each hot dog is sold for $2.25 and makes a profit of $0.75. Hot dogs left over after the game are sold to the student cafeteria for $0.50 each. Based on previous games, the demand is normally distributed with an average of 2000 hot dogs sold per game and a standard deviation of 400. Find the optimal stocking level for hot dogs.

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