Chapter 15: Inventory Systems For Independent Demand

Definition of Inventory-Stock of any item or resource used in the

organization.Inventory System – Set of policies and controls that

monitors levels of inventory, determines what level should be maintained, when stock should be replenished, , and how large orders should be.

Manufacturing inventory – Raw materials, finished products, component parts, supplies, and work in process (WIP) 1

Basic Purpose of Inventory Analysis• -To determine (a) when item should be

ordered, and (b) how large the order should be.

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Purposes of Inventory• To maintain independence of operations – A

supply of materials at a work center allows that center flexibility in operations. Independence of workstations is desirable on assembly lines as well. The time that it takes to do identical operations will naturally vary from one unit to another. Therefore, it is desirable to have a cushion of several parts within the workstation so that shorter performance times can compensate for longer performance times.

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Purposes of Inventory (Continued)• To variation in product demand – Demand is not

completely known. Therefore, a safety or buffer stock must be maintained to absorb variation.

• To allow flexibility in production scheduling – A stock of inventory relieves the pressure on the production system to get the goods out. This causes longer lead times which permit production planning for smoother flow lower-cost operation through larger lot-size production. 4

Purposes of Inventory (Continued)• To provide a safeguard for variation in raw

material delivery time – When material is ordered from a vendor, delays can occur for a variety of reasons: a normal variation in shipping time, a shortage of material at the vendor’s plant causing backlog, unexpected strike at the vendor’s plant or at one of the shipping companies etc. Inventory will play as a safeguard.

• To take advantage of economic purchase-order size – There are costs to place an order: labor, communication cost, handling cost, etc. The larger the order, the lower the per-unit cost.

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Inventory Costs• Holding (or carrying cost): costs for storage

facilities, handling, insurance, pilferage, breakage, obsolescence, depreciation, taxes, opportunity cost of the capital, etc.

• Set-up (or production change) costs: Costs to obtain necessary materials, arranging specific equipments set-up, filling-out the required papers, appropriately charging time and materials, moving out the previous stock of material, etc.

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Inventory Costs (Continued)• Ordering costs: Managerial and clerical costs to

prepare the purchase or production order

• Shortage costs: Costs for unfilled demand, lost customers effect, etc.

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Inventory Models- Fixed-order quantity models (economic order quantity

model, Q-model), and Fixed-time period models (periodic system, periodic review system, fixed-order interval system, P-model)

- Fixed-order quantity models are “event triggered” and fixed-time period models are “time triggered”

- A fixed-order quantity model initiates an order when the event of reaching a specified reorder level occurs. In contrast, a fixed-time period model initiate an order after a predetermined time elapses.

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Fixed-order vs. Fixed-time• The fixed-time period model has a larger average inventory

because it must also protect against stock out during the review period, T: the fixed-order quantity model has no review period.

• The fixed-order quantity model favors more expensive items because average inventory is lower.

• The fixed-order quantity model is more appropriate for important items such as critical repair parts because there is closer monitoring and therefore quicker response to potential stock out.

• The fixed-order quantity model requires more time to maintain because every addition or withdrawal is logged.9

Fixed-Order Quantity Models- Determine the specific point, R, at which an

order will be placed and the size of that order, Q.

- R is always a specified number of units. An order of size Q is placed when the inventory available (currently available and on order) reaches the point R.

Inventory position = On-hand + On-order – Back-ordered

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Assumptions

• Demand for the product is constant and uniform throughout the period

• Lead time (time from ordering to receipt) is constant

• Price per unit of product is constant

• Inventory holding cost is based on average inventory

• Ordering or setup costs are constant

• All demands for the product will be satisfied (No back orders are allowed)

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BASIC FIXED-ORDER QUANTITY MODEL AND REORDER POINT BEHAVIOR

R = Reorder pointQ = Economic order quantityL = Lead time

L L

Q QQ

R

Time

Numberof unitson hand

1. You receive an order quantity Q.

2. Your start using them up over time. 3. When you reach down to a

level of inventory of R, you place your next Q sized order.

4. The cycle then repeats.

COST MINIMIZATION GOAL

Ordering Costs

HoldingCosts

Order Quantity (Q)

COST

Annual Cost ofItems (DC)

Total Cost

QOPT

By adding the item, holding, and ordering costs together, we determine the total cost curve, which in turn is used to find the Qopt inventory order point that minimizes total costs

By adding the item, holding, and ordering costs together, we determine the total cost curve, which in turn is used to find the Qopt inventory order point that minimizes total costs

BASIC FIXED-ORDER QUANTITY (EOQ) MODEL FORMULA

H 2

Q + S

Q

D + DC = TC

Total Annual =Cost

AnnualPurchase

Cost

AnnualOrdering

Cost

AnnualHolding

Cost+ +

TC=Total annual costD =DemandC =Cost per unitQ =Order quantityS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding and storage cost per unit of inventory

TC=Total annual costD =DemandC =Cost per unitQ =Order quantityS =Cost of placing an order or setup costR =Reorder pointL =Lead timeH=Annual holding and storage cost per unit of inventory

DERIVING THE EOQUsing calculus, we take the first

derivative of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Qopt

Using calculus, we take the first derivative of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Qopt

Q = 2DS

H =

2(Annual D em and)(Order or Setup Cost)

Annual Holding CostOPT

Reorder point, R = d L_

d = average daily demand (constant)

L = Lead time (constant)

_

We also need a reorder point to tell us when to place an order

We also need a reorder point to tell us when to place an order

EOQ EXAMPLE (1) PROBLEM DATA

Annual Demand = 1,000 unitsDays per year considered in average

daily demand = 365Cost to place an order = $10Holding cost per unit per year = $2.50Lead time = 7 daysCost per unit = $15

Given the information below, what are the EOQ and reorder point?Given the information below, what are the EOQ and reorder point?

EOQ EXAMPLE (1) SOLUTION

Q = 2DS

H =

2(1,000 )(10)

2.50 = 89.443 units or OPT 90 units

d = 1,000 units / year

365 days / year = 2.74 units / day

Reorder point, R = d L = 2.74units / day (7days) = 19.18 or _

20 units

In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 units.

In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, when you only have 20 units left, place the next order of 90 units.

EOQ EXAMPLE (2) PROBLEM DATA

Annual Demand = 10,000 unitsDays per year considered in average daily demand = 365Cost to place an order = $10Holding cost per unit per year = 10% of cost per unitLead time = 10 daysCost per unit = $15

Determine the economic order quantity and the reorder point given the following…

Determine the economic order quantity and the reorder point given the following…

EOQ EXAMPLE (2) SOLUTION

Q =2DS

H=

2(10,000 )(10)

1.50= 365.148 units, or OPT 366 units

d =10,000 units / year

365 days / year= 27.397 units / day

R = d L = 27.397 units / day (10 days) = 273.97 or _

274 units

Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.

Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.

Fixed-Order Quantity Model With Usage During Production Time

• The previous model assumed that the quantity ordered would be received in one lot, but frequently this is not the case.

• In many situations, production of an inventory item and usage of that item take place simultaneously (where one part of a production system acts as supply to another part).

• Also, companies are beginning to longer-term arrangements with supplier. Under such contracts, a single order may cover product or material needs over a six-month or year period, with the vendor making deliveries weekly or sometimes more frequently.

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Fixed-Order Quantity Model With Usage During Production Time (Continued)

• TC = DC + (D/Q)*S + ((p-d)/2p)*QH

p = Production rate, d = Daily demand rate

After differentiating with respect to Q and setting the equation to 0,

Qopt =

21

)*2

(dp

p

H

DS

Standard Deviation• Standard deviation, σ, is a measurement of deviation

from mean. In case of demand, it is a measurement of deviation from average demand.

σd =standard deviation for daily demand=

If lead time is L, then σL =

22

n

ddn

i i

1

2)(

2d

L

Establishing Safety Stock Levels• Up to now, the assumption is that demand is constant and

is precisely known.• On the contrary, in the majority of the case demand is not

constant but varies from day to day. Hence, safety stock must be maintained to provide some level of protection against stock-outs.

Safety stock can be defined as the amount of inventory carried in addition to the expected demand.

If demand is assumed to follow normal distribution, then the expected demand is the mean.

If average weekly demand is 100 units and the demand for next week is expected to be same, then if 120 units is carried in inventory, then safety stock is (120 – 100) = 20 units.

Service Level• Service level refers to the number of units demanded

that can be supplied from stock currently on hand.

For example, if annual demand for an item is 1000 units, a 95% service level means that 950 units can be supplied immediately from stock and 50 units are short

The discussion on service levels is based on a statistical concept known as Expected z or E(z). E(z) is the expected number of units short during each lead time (assumption is that demand is normally distributed)

Service Level (Continued)• For example, assume that the average weekly demand for

an item is 100 units with a standard deviation of 10 units. If there is 110 units in the inventory at the beginning of the week, how many items will be short?

• If the demand comes out to be from 111 units to

the number of units short is 1, 2, 3,…….,

respectively.

Expected number of short = 1*P(demand = 111 units) + 2*P(demand = 112 units) + 3*P(demand = 113 units) + ……….+ *P(demand = α )

Service Level• If z = 1 and standard deviation, σL = 10 units, then

amount of safety stock = z σL = 1*10=10 units

Then, expected number of short is E(z)* σL [E(z) have to be determined from Exhibit 15.6 for value of z]

Service level = (1 – E(z)* σL /Inventory or Ordered Amount)*100%

Service Level Example 1• Consider an economic order quantity case where

annual demand D = 1000 units, economic order quantity Qopt = 200 units, the desired service level P = 0.95, the standard deviation of demand during lead time σL = 25 units, and lead time L = 15 days. Determine the reorder point.

Service Level Example 2• Daily demand for a certain product is normally

distributed with a mean of 60 and standard deviation of 7. The source of supply is reliable and maintains a constant lead time of 6 days. The cost of placing an order is $10 and annual holding costs are $0.50 per unit. There are no stock-out costs, and unfilled orders are filled as soon as the order arrives. Assume sales occur over the entire year. Find the economic order quantity and reorder point to satisfy 95% of the customers from stock on hand.

Fixed Time Period Models• In a fixed time period system, inventory is counted

only at particular times, such as every week or every month. Counting inventory and placing orders on a periodic basis is desirable in situations such as when vendors make routine visits to customers and take orders for their complete line of orders.

• Fixed-time period model generate order quantities that vary from period to period depending on usage rates. These generally require a higher level of safety stock than a fixed-order quantity system. It is possible that some large demand will draw the stock down to 0 right after an order is placed. This condition could go unnoticed until the next review period.

Fixed-Time Period Model With Specified Service Level

• In a fixed –time period system, reorders are placed at the time of review (T), and the safety stock that must be reordered is

Safety Stock = zσT+L

The quantity to order, q , is

Order quantity = Average demand over the vulnerable period + Safety Stock – Inventory Currently On Hand (Plus On Order, If Any)

q = d(T+L) +̄ zσT+L – I

E(z) = T(1-P)/d̄ σT+L

EXAMPLE OF THE FIXED-TIME PERIOD MODEL

Average daily demand for a product is 20 units. The review period is 30 days, and lead time is 10 days. Management has set a policy of satisfying 96 percent of demand from items in stock. At the beginning of the review period there are 200 units in inventory. The daily demand standard deviation is 4 units.

Given the information below, how many units should be ordered?Given the information below, how many units should be ordered?

EXAMPLE OF THE FIXED-TIME PERIOD MODEL: SOLUTION (PART 1)

T+ L d2 2 = (T + L) = 30 + 10 4 = 25.298

E(z) = d T(1-P)/̄ σT+L

= 20*30(1-0.96)/25.298

= 0.949

If E(z) = 0.949, then z = -0.84

[From Exihibit 15.6, for E(z) = 1.000, z = -0.90, for E(z) =

0.920, z = -0.80

So, for E(z) = 0.949, z = -0.90 + {(1.000-0.949)/(1.000-

0.920)}*{-0.80 –(-0.90)}=-0.90+(.051/.08)*.10 =-0.84]

E(z) = d T(1-P)/̄ σT+L

= 20*30(1-0.96)/25.298

= 0.949

If E(z) = 0.949, then z = -0.84

[From Exihibit 15.6, for E(z) = 1.000, z = -0.90, for E(z) =

0.920, z = -0.80

So, for E(z) = 0.949, z = -0.90 + {(1.000-0.949)/(1.000-

0.920)}*{-0.80 –(-0.90)}=-0.90+(.051/.08)*.10 =-0.84]

Fixed-Time Period Example (Continued)The quantity to order = d (T+L)+z̄ σT+L –I

=20(30+10) + (-0.84)*25.298 – 200

=578.75 or 579 units

Background for ABC Inventory PlanningMaintaining inventory through counting, placing orders,

receiving stocks, placing the units in the right place, and so on takes personnel time and costs money.

When there are limits on these resources, the logical move is to try to use the available resources to control inventory in the best way.

In other words, focus on the most important items in inventory.

Pareto Principle• In the 19th Century, Villefredo Pareto, in a study of the

distribution of wealth in Milan, found that 20% of the people controlled 80% of the wealth.

• This logic of the few having the greatest importance and many having little importance has been broadened to include many situations and is termed the Pareto Principle.

Pareto Principle (Continued)• Pareto principle is true in our everyday lives and is

certainly true in inventory system (where a few items account for the bulk of investment).

• Any inventory system must specify when an order is to be placed for an item and how many units to order. Most inventory control systems involve so many items that is not practical to model and give thorough treatment to each item.

• To resolve this, the ABC classification scheme divides inventory items into three different groupings: high dollar volume (A), moderate dollar volume (B), and low dollar volume (C).

ABC Classification• If the annual usage of items in inventory is listed

according to dollar volume, generally the list shows that a small number of items account for a large dollar volume and that a large number of items account for a small dollar volume.

• The ABC approach divides the list into three groupings by value: A items constitute roughly the top 15 percent of the items, B items the next 35 percent, and C items the last 50 percent. A items account for roughly 70 percent of the dollar volume, B items account for around 20 percent of the dollar volume, and C items account for close to 10 percent of the total dollar volume.

ABC Classification (Continued)• The values are not exactly fixed. The objective is to

separate the important from unimportant.• The purpose of classifying items into groups is to

establish the appropriate degree of control over each item.