1 introduction to operations management inventory management chapte r 11

1

Introduction to Operations Management

Inventory Management

CHAPTER

11

2


What is inventory?

An inventory is an idle stock of material used to facilitate production or to satisfy customer needs

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Why do we need inventory?

Economies of scales in ordering or production - cycle stock– trade-off setup cost vs carrying cost

Smooth production when requirements have predictable variability - seasonal stock– trade-off production adjustment cost vs carrying

cost

4


Why do we need inventory? (cont’d)

Provide immediate service when requirements are uncertain (unpredicatable variability) -safety stock– trade-off shortage cost versus carrying cost

Decoupling of stages in production/distribution systems -decoupling stock– trade-off interdependence between stages vs

carrying cost

5


Why do we need inventory? (cont’d)

Production is not instantaneous, there is always material either being processed or in transit - pipeline stock– trade-off cost of speed vs carrying cost

6


Why carrying inventory

As you can see from the above list, there are economic reasons, technological reasons as well as management/operations reasons.

7


Why carrying inventory is costly?

Traditionally, carrying costs are attributed to:– Opportunity cost (financial cost)– Physical cost

(storage/handling/insurance/theft/obsolescence/…)

8


Inventory Management

A

B(4) C(2)

D(2) E(1) D(3) F(2)

Independent Demand

Dependent Demand

Independent demand is uncertain. Dependent demand is certain.

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Types of Inventories

Raw materials & purchased parts Partially completed goods called

work in progress Finished-goods inventories

– (manufacturing firms) or merchandise (retail stores)

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Types of Inventories (Cont’d)

Replacement parts, tools, & supplies Goods-in-transit to warehouses or

customers

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Inventory Counting Systems

Periodic SystemPhysical count of items made at periodic intervals

Perpetual Inventory System System that keeps track of removals from inventory continuously, thus monitoringcurrent levels of each item

12


Inventory Counting Systems (Cont’d)

Two-Bin System - Two containers of inventory; reorder when the first is empty

Universal Bar Code - Bar code printed on a label that hasinformation about the item to which it is attached

0

214800 232087768

13


ABC Classification System

Classifying inventory according to some measure of importance and allocating control efforts accordingly.

A - very important

B - mod. important

C - least importantA

BC

High

Annual $ volume of items

Low

Few ManyNumber of Items

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All models are trying to answer the following questions for given informational, economic and technological characteristics of the operating environment: – How much to order (produce)?– When to order (produce)?– How often to review inventory?– Where to place/position inventory?

What are we tackling in inventory management?

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The inventory models

The quantitative inventory management models range from simple to very complicated ones. However, there are two simple models that capture the essential tradeoffs in inventory theory– The newsboy (newsvendor) model– The EOQ (Economic Ordering Quantity) model

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The Newsboy Model

The newsboy model is a single-period stocking problem with uncertain demand: Choose stock level and then observe actual demand.– Tradeoff: overstock cost versus opportunity

cost (lost of profit because of under stocking)

17


The Newsboy Model

Shortage cost (Cs) - the opportunity cost for lost of sales as well as the cost of losing customer goodwill. Cs = revenue per unit - cost per unit

Excess cost (Ce) - the over-stocking cost (the cost per item not being able to sell) Ce = Cost per unit - salvage value per unit

18


The Newsboy Model

In this model, we have to consider two factors (decision variables): the supply (X), also know as the stock level, and the demand (Y). The supply is a controllable variable in this case and the demand is not in our control. We need to determine the quantity to order so that long-run expected cost (excess and shortage) is minimized.

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The Newsboy Model

Let X = n and P(n) = P(Y>n). The question is: Based on what should we stock this n-th unit?

The opportunity cost (expected loss of profit)for this n-th unit is P(n) Cs

The expected cost for not being able to sell this n-th unit (expected loss) (1-P(n))Ce

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The Newsboy model

Stock the n-th unit if P(n)Cs > (1-P(n))Ce

Do not stock if P(n)Cs < (1-P(n))Ce

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The Newsboy model

The equilibrium point occurs at P(n)Cs = (1-P(n)) Ce

Solving the equation P(n) = Ce/(Cs + Ce)

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Newsboy model

Service level is the probability that demand will not exceed the stocking level and is the key to determine the optimal stocking level.

In our notation, it is the probability that YX(=n), which is given by

P(Yn) = 1 - P(n) = Cs/(Cs + Ce)

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Example (p.564)

Demand for long-stemmed red roses at a small flower shop can be approximated using a Poisson distribution that has a mean of four dozen per day. Profit on the roses is $3 per dozen. Leftover are marked down and sold the next day at a loss of $2 per dozen. Assume that all marked down flowers are sold. What is the optimal stocking level?

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Example (solutions)

Cs = $3, Ce = $2

Opportunity cost is P(n)Cs Expected loss for over-

stocking (1 - P(n)) Ce

P(Y n) = Cs/(Cs + Ce) = 3/((3+2) = 0.6

From the table, it is between 3 and 4, round up give you optimal stock of 4 dozen.

Demand (dznper day)

Cumulativefrequency

0 0.018

1 0.092

2 0.238

3 0.434

4 0.629

5 0.7855

Cumulative frequencies for Poisson distribution, mean = 4

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The Inventory CycleProfile of Inventory Level Over Time

Quantityon hand

Q

Receive order

Placeorder

Receive order

Placeorder

Receive order

Lead time

Reorderpoint

Usage rate

Time

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How to estimate the inventory costs?

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Estimating the inventory costs

Q = quantity to order in each cycle S = fixed cost or set up cost for each order D = demand rate or demand per unit time H = holding cost per unit inventory per unit

time c = unit cost of the commodity

TC = total holding cost per unit time

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Estimating the inventory cost

The cost per order = S + cQ

The length of order cycle = Q/D unit time

The average inventory level = Q/2 Thus the inventory carrying cost

= (Q/2) H (Q/D) = HQ2/(2D)

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Estimating the inventory cost

The total cost per order cycle = S + cQ + HQ2/(2D)

Thus the total cost per unit time is dividing the above expression by Q/D, I.e.,

TC = {S + cQ + HQ2/(2D)} {D/Q} = DS/Q + cD + HQ/2

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Total Cost

Annualcarryingcost

Annualorderingcost

Total cost = +

Q2

H DQ

STC = +

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Deriving the EOQ

Using calculus, we take the derivative of the total cost function and set the derivative (slope) equal to zero and solve for Q.

Q = 2DS

H =

2(Annual Demand)(Order or Setup Cost)

Annual Holding CostOPT

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Cost Minimization GoalThe Total-Cost Curve is U-Shaped

Ordering Costs

QO Order Quantity (Q)

Ann

ual C

ost

(optimal order quantity)

TCQH

D

QS

2

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Minimum Total Cost

The total cost curve reaches its minimum where the carrying and ordering costs are equal.

Q = 2DS

H =

2(Annual Demand)(Order or Setup Cost)

Annual Holding CostOPT

34


Total CostsC

ost

EOQ

TC with PD

TC without PD

PD

0 Quantity

Adding Purchasing costdoesn’t change EOQ

35


Total Cost with Constant Carrying Costs

OC

EOQ Quantity

Tot

al C

ost

TCa

TCc

TCb

Decreasing Price

CC a,b,c

36


Example (Example 2, p.541)

A local distributor for a national tire company expects to sell approximately 9600 steel-belted radial tires of a certain size and tread design next year. Annual carrying costs are $16 per tire, and ordering costs are $75. This distributor operate 288 days a year.

a. What is the EOQ?

b. How many times per year does the store reorder?

c. What is the length or an order cycle?

d. What is the total ordering and inventory cost?

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Solution (Example 2, p.541)

D = 9600 tires per year, H = $16 per unit per year, S = $75

a. Q0 = {2DS/H} = {2(9600)75/16} = 300

b. Number of orders per year = D/ Q0 = 9600 / 300 = 32

c. The length of one order cycle = 1 / 32 years = 288/32 days = 9 days

d. Total ordering and inventory cost = QH/2 + DS/Q

= 2400 + 2400 = 4800

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EOQ with quantity discount

This is a variant of the EOQ model. Quantity discount is another form of economies of scale: pay less for each unit if you order more. The essential trade-off is between economies of scale and carrying cost.

39


EOQ with quantity discount

To tackle the problem, there will be a separate (TC) curve for each discount quantity price. The objective is to identify an order quantity that will represent the lowest total cost for the entire set of curves in which the solution is feasible. There are two general cases:– The holding cost is constant– The holding cost is a percentage of the

purchasing price.

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Quantity discount (constant holding cost)

Total cost per cycle = setup cost + purchase cost + carrying cost

Setup cost = S Purchase cost = ciQ, if the order quantity Q

is in the range where unit cost is ci. Carrying cost = (Q/2)h(Q/D)

TC = {S+ciQ+hQ2/(2D)}{D/Q} (annual cost)

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h

2DS Q0

Differentiating, we obtain an optimal order quantity which is independent of the price of the good

The questions is: Is this Q0 a feasible solution to our problem?

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Solution steps: 1. Compute the EOQ. 2. If the feasible EOQ is on the lowest price

curve, then it is the optimal order quantity. 3. If the feasible EOQ is on other curve, find

the total cost for this EOQ and the total costs for the break points of all the lower cost curves. Compare these total costs. The point (EOQ point or break point) that yields the lowest total cost is the optimal order quantity.

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Example

The maintenance department of a large hospital uses about 816 cases of liquid cleanser annually. Ordering costs are $12, carrying cost are $4 per case a year, and the new price schedule indicates that orders of less than 50 cases will cost $20 per case, 50 to 79 cases will cost $18 per case, 80 to 99 cases will cost $17 per case, and larger orders will cost $16 per case. Determine the optimal order quantity and the total cost.

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Example (solutions) 1. The common EOQ: = {2(816)(12)/4} = 70

2. 70 falls in the range of 50 to 79, at $18 per case. TC = DS/Q+ciD+hQ/2

= 816(12)/70 + 18(816) + 4(70)/2 = 14,968

3. Total cost at 80 cases per order TC = 816(12)/80 + 17(816) + 4(80)/2 = 14,154

Total cost at 100 cases per order TC = 816(12)/100 + 16(816) + 4(100)/2 =13,354

The minimum occurs at the break point 100. Thus order 100 cases each time

45


Quantity discount (h = rc)

Using the same argument as in the constant holding cost case, Let TCi be the total cost when the unit quantity is ci. TCi = rciQ/2 + DS/Q + ciD

Therefore EOQ = {2DS/(rci)}

In this case, the carrying cost will decrease with the unit price. Thus the EOQ will shift to the right as the unit price decreases.

46


Quantity discount (h = rc)

Steps to identify optimal order quantity

1. Beginning with the lowest price, find the EOQs for each price range until a feasible EOQ is found.

2. If the EOQ for the lowest price is feasible, then it is the Optimal order quantity.

3. Same as step (3) in the constant carrying case.

47


Example (p.549)

Surge Electric uses 4000 toggle switches a year. Switches are priced as follows: 1 to 499 at $0.9 each; 500 to 999 at $.85 each; and 1000 or more will be at $0.82 each. It costs approximately $18 to prepare an order and receive it. Carrying cost is 18% of purchased price per unit on an annual basis. Determine the optimal order quantity and the total annual cost.

48


SolutionD = 4000 per year; S = 18 ; h = 0.18 {price}

Step 1. Find the EOQ for each price, starting with the lowest price

EOQ(0.82) = {2(4000)(18)/[(0.18)(0.82)]} = 988 (Not feasible for the price range).

EOQ(0.85)= 970 (feasible for the range 500 to 999)

Step 2. Feasible solution is not on the lowest cost curve

Step 3. TC(970) = 970(.18)(.85)/2 + 4000(18)/970 + .85(4000) = 3548

TC(1000) = 1000(.18)(.82)/2 + 4000(18)/1000 + .82(4000) = 3426

Thus the minimum total cost is 3426 and the minimum cost order size is 1000 units per order.

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We have settled the problem of how much to order. The next decision problem is when to order.

50


Reorder point

In the EOQ model, we assume that the delivery of goods is instantaneous. However, in real life situation, there is a time lag between the time an order is placed and the receiving of the ordered goods. We call this period of time the lead time. Demand is still occurring during the lead time and thus inventory is required to meet customer demand. This is why we need to consider when to order!

51


When to reorder

We can reorder based on:– time, say weekly, monthly, etc– quantity of goods on hand

52


When to Reorder with EOQ Ordering

Reorder Point - When the quantity on hand of an item drops to this amount, the item is reordered

Safety Stock - Stock that is held in excess of expected demand due to variable demand rate and/or lead time.

Service Level - Probability that demand will not exceed supply during lead time.

53


Safety Stock

LT Time

Expected demandduring lead time

Maximum probable demandduring lead time

ROP

Qu

ant

ity

Safety stock

54


When to reorder

In general, we need to consider the following four factors when deciding when to order:– The rate of demand (usually based on a

forecast)– The length of lead time.– The extent of demand and lead time variability.– The degree of stock-out risk acceptable to

management.

55


When to reorder

We will just cover two simple cases:– The demand rate and lead time are

constant– Variable demand rate and constant lead

time.

56


Reorder Point

ROP

Risk ofa stockout

Service level

Probability ofno stockout

Expecteddemand Safety

stock0 z

Quantity

z-scale

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ROP - constant demand and lead time

Let d be the demand rate per unit time and LT be the lead time in unit time.

The reorder point quantity is given by ROP = expected demand during lead time +

safety stock during lead time In this case, safety stock required is 0

Thus ROP = d (LT)

Example: See Example 7 on page 551

58


ROP - variable demand rate

The lead time is constant. In this case, we need extra buffer of safety stock to insure against stock-out risk.

Since it cost money to carry safety stocks, a manager must weigh carefully the cost of carrying safety stock against the reduction in stock-out risk (provided by the safety stock)

59


ROP - variable demand rate

In other words, he needs to trade-off the cost of safety stock and the service level.

Service level = 1 - stock-out risk ROP = Expected lead time demand + safety stock

= d (LT) + z Where is the standard deviation of the lead time

demand = {LT} d

Therefore ROP = d(LT) + z {LT} d

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Example (Problem 4, p.569)The housekeeping department of a hotel uses approximately 400 washcloths per day. The actual amount tends to vary with the number of guests on any given night. Usage can be approximated by a normal distribution that has a mean of 400 and a standard deviation of 9 washcloths per day. A linen supply company delivers towels and washcloths with a lead time of three days. If the hotel policy is to maintain a stock-out risk of 2 percent, what is the minimum number of washcloths that must be on hand at reorder time, and how much of that amount can be considered safety stock?

Solution:

d = 400, LT = 3 days, d = 9 , service level = 1 - risk = 0.98

61


Example (solution)

Z = 2.055 (from normal distribution table)

Thus ROP = 400 (3) + 2.055 (3) d = 1200 + 32.03 = 1232

The safety stock is approximately 32 washcloths, providing a service level of 98%.

62


Fixed-order-interval model

In the EOQ/ROP models, fixed quantities of items are ordered at varying time interval. However, many companies ordered at fixed intervals: weekly, biweekly, monthly, etc. They order varying quantity at fixed intervals. We called this class of decision models fixed-order-interval (FOI) model.

63


FOI model

Why use FOI model?

Is it optimal?

What are decision variables in the FOI model?

64


FOI model

Assuming lead time is constant, we need to consider the following factors– the expected demand during the ordering

interval and the lead time– the safety stock for the ordering interval and

lead time– the amount of inventory on hand at the time

of ordering

65


FOI model

OILT

Receive order

Safety stock

Time

Am

ount

on

hand

Place order

Order quantity

Amount on hand (A)

66


FOI model

We use the following notations OI = Length of order interval A = amount of inventory on hand LT = lead time d = Standard deviation of demand

67


FOI model

Amount to order = expected demand during protection interval

+ Safety stock for protection interval - amount on hand at time of placing order

= ALTOIz LT) d(OI d

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Example (p.560)

The following information is given for a FOI system. Determine the amount to order.

d = 30 units per day, d = 3 units per day, LT = 2 days; OI = 7 days

Amount on hand = 71 units, Desired serve level = 99%.

Solution: z = 2.33 for 99% service level.

Amount to order = 30 (7 + 2) + 2.33 (3) (7 + 2) - 71 = 270 + 20.97 - 71 = 220

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