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5 Reorder Point: Determining When to Order

5 Reorder Point: Determining When to Order

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206



CHAPTER 6 • INVENTORY CONTROL MODELS



FIGURE 6.4

Reorder Point Graphs



Inventory

Level

Q



ROP

0

Lead time = L

ROP < Q



Time



Inventory

Level

On Order

Q



On hand



0

Lead time = L

ROP > Q



Time



Since the maximum on-hand inventory level is the order quantity of 400, an inventory position

of 480 would be:

Inventory position = 1Inventory on hand2 + 1Inventory on order2

480 = 80 + 400

Thus, a new order would have to be placed when the on-hand inventory fell to 80 while there

was one other order in-transit. The second graph in Figure 6.4 illustrates this type of situation.



6.6



EOQ Without the Instantaneous Receipt Assumption



The production run model

eliminates the instantaneous

receipt assumption.



When a firm receives its inventory over a period of time, a new model is needed that does not

require the instantaneous inventory receipt assumption. This new model is applicable when

inventory continuously flows or builds up over a period of time after an order has been placed or

when units are produced and sold simultaneously. Under these circumstances, the daily demand

rate must be taken into account. Figure 6.5 shows inventory levels as a function of time. Because

this model is especially suited to the production environment, it is commonly called the

production run model.

In the production process, instead of having an ordering cost, there will be a setup cost. This

is the cost of setting up the production facility to manufacture the desired product. It normally

includes the salaries and wages of employees who are responsible for setting up the equipment,

engineering and design costs of making the setup, paperwork, supplies, utilities, and so on. The

carrying cost per unit is composed of the same factors as the traditional EOQ model, although

the annual carrying cost equation changes due to a change in average inventory.



6.6



FIGURE 6.5

Inventory Control and

the Production Process



Inventory

Level



EOQ WITHOUT THE INSTANTANEOUS RECEIPT ASSUMPTION



Part of Inventory Cycle

During Which Production Is

Taking Place



207



There Is No Production

During This Part of the

Inventory Cycle



Maximum

Inventory



t



Solving the production run

model involves setting setup costs

equal to holding costs and solving

for Q.



Time



The optimal production quantity can be derived by setting setup costs equal to holding

or carrying costs and solving for the order quantity. Let’s start by developing the expression

for carrying cost. You should note, however, that making setup cost equal to carrying cost

does not always guarantee optimal solutions for models more complex than the production

run model.



Annual Carrying Cost for Production Run Model

As with the EOQ model, the carrying costs of the production run model are based on the average inventory, and the average inventory is one-half the maximum inventory level. However,

since the replenishment of inventory occurs over a period of time and demand continues during

this time, the maximum inventory will be less than the order quantity Q. We can develop the

annual carrying, or holding, cost expression using the following variables:

Q = number of pieces per order, or production run

Cs = setup cost

Ch = holding or carrying cost per unit per year

p = daily production rate

d = daily demand rate

t = length of production run in days

The maximum inventory level is as follows:

1Total produced during the production run2 - 1Total used during production run2

= 1Daily production rate21Number of days of production2

-1Daily demand21Number of days of production2



= 1pt2 - 1dt2

Since



total produced = Q = pt,

we know that

t =



Q

p



Maximum inventory level = pt - dt = p

The maximum inventory level

in the production model is less

than Q.



Q

Q

d

- d = Qa1 - b

p

p

p



Since the average inventory is one-half of the maximum, we have

Average inventory =



Q

d

a1 - b

p

2



(6-9)



208



CHAPTER 6 • INVENTORY CONTROL MODELS



and

Annual holding cost =



Q

d

a 1 - bCh

p

2



(6-10)



Annual Setup Cost or Annual Ordering Cost

When a product is produced over time, setup cost replaces ordering cost. Both of these are independent of the size of the order and the size of the production run. This cost is simply the number of orders (or production runs) times the ordering cost (setup cost). Thus,

Annual setup cost =



D

Cs

Q



(6-11)



Annual ordering cost =



D

C

Q o



(6-12)



and



Determining the Optimal Production Quantity

When the assumptions of the production run model are met, costs are minimized when the setup

cost equals the holding cost. We can find the optimal quantity by setting these costs equal and

solving for Q. Thus,

Annual holding cost = Annual setup cost

Q

d

D

a1 - bCh =

C

p

2

Q s

Here is the formula for the

optimal production quantity.

Notice the similarity to the basic

EOQ Model.



Solving this for Q, we get the optimal production quantity 1Q*2:

Q* =



2DCs



(6-13)



d

Ch a1 - b

p

Q



It should be noted that if the situation does not involve production but rather involves the receipt

of inventory over a period of time, this same model is appropriate, but Co replaces Cs in the

formula.

Production Run Model

Annual holding cost =

Annual setup cost =



Q

d

a1 - bCh

p

2

D

C

Q s

2DCs



Optimal production quantity Q * =

Q



Ch a1 -



d

b

p



Brown Manufacturing Example

Brown Manufacturing produces commercial refrigeration units in batches. The firm’s estimated

demand for the year is 10,000 units. It costs about $100 to set up the manufacturing process, and

the carrying cost is about 50 cents per unit per year. When the production process has been

set up, 80 refrigeration units can be manufactured daily. The demand during the production

period has traditionally been 60 units each day. Brown operates its refrigeration unit production

area 167 days per year. How many refrigeration units should Brown Manufacturing produce in



6.6



EOQ WITHOUT THE INSTANTANEOUS RECEIPT ASSUMPTION



209



each batch? How long should the production part of the cycle shown in Figure 6.5 last? Here is

the solution:

Annual demand = D = 10,000 units

Setup cost = Cs = $100

Carrying cost = Ch = $0.50 per unit per year

Daily production rate = p = 80 units daily

Daily demand rate = d = 60 units daily

2DCs



1. Q * =

R

2. Q * =



=



Ch a1 -



d

b

p



2 * 10,000 * 100

60

b

0.5a1 80

R

2,000,000



B 0.5 A 1>4 B



= 116,000,000



= 4,000 units

If Q* = 4,000 units and we know that 80 units can be produced daily, the length of each production cycle will be Q>p = 4,000>80 = 50 days. Thus, when Brown decides to produce refrigeration units, the equipment will be set up to manufacture the units for a 50-day time span.

The number of production runs per year will be D>Q = 10,000>4,000 = 2.5. This means that

the average number of production runs per year is 2.5. There will be 3 production runs in one

year with some inventory carried to the next year, so only 2 production runs are needed in the

second year.

USING EXCEL QM FOR PRODUCTION RUN MODELS The Brown Manufacturing production run



model can also be solved using Excel QM. Program 6.2A contains the input data and the Excel

formulas for this problem. Program 6.2B provides the solution results, including the optimal production quantity, maximum inventory level, average inventory level, and the number of setups.



PROGRAM 6.2A



Excel QM Formulas and Input Data for the Brown Manufacturing Problem



Enter the demand rate, setup cost, and

holding cost. Notice that the holding cost

is a fixed dollar amount rather than a

percentage of the unit price.

Enter daily production rate

and daily demand rate.



Calculate the optimal production quantity.



Calculate the

average number

of setups.



Calculate the maximum inventory.



Calculate the annual holding costs based on average inventory

and the annual setup cost based on the number of setups.



210



CHAPTER 6 • INVENTORY CONTROL MODELS



PROGRAM 6.2B

The Solution Results

for the Brown

Manufacturing Problem

Using Excel QM



IN ACTION



Fortune 100 Firm Improves Inventory

Policy for Service Vehicles



M



ost manufacturers of home appliances provide in-home

repair of the appliances that are under warranty. One such Fortune 100 firm had about 70,000 different parts that were used in

the repair of its appliances. The annual value of the parts inventory was over $7 million. The company had more than 1,300 service vehicles that were dispatched when service requests were

received. Due to limited space on these vehicles, only about 400

parts were typically carried on each one. If a service person

arrived to repair an appliance and did not have necessary part

(i.e., a stockout occured), a special order was made to have the

order delivered by air so that the repair person could return and

fix the appliance as soon as possible.

Deciding which parts to carry was a particularly difficult

problem. A project was begun to find a better way to forecast the

demand for parts and identify which parts should be stocked on

each vehicle. Initially, the intent was to reduce the parts inventory

on each truck as this represented a significant cost of holding



6.7



the inventory. However, upon further analysis, it was decided that

the goal should be to minimize the overall cost—including the

costs of special deliveries of parts, revisiting the customer for

the repair if the part was not initially available, and overall customer satisfaction.

The project team improved the forecasting system used to

project the number of parts needed on each vehicle. As a result,

the actual number of parts carried on each vehicle increased.

However, the number of first-visit repairs increased from 86% to

90%. This resulted in a savings of $3 million per year in the cost

of these repairs. It also improved customer satisfaction because

the problem was fixed without the service person having to

return a second time.

Source: Based on Michael F. Gorman and Sanjay Ahire. “A Major Appliance

Manufacturer Rethinks Its Inventory Policies for Service Vehicles,” Interfaces

36, 5 (September–October 2006): 407–419.



Quantity Discount Models

In developing the EOQ model, we assumed that quantity discounts were not available. However,

many companies do offer quantity discounts. If such a discount is possible, but all of the other

EOQ assumptions are met, it is possible to find the quantity that minimizes the total inventory

cost by using the EOQ model and making some adjustments.



6.7



TABLE 6.3

Quantity Discount

Schedule



QUANTITY DISCOUNT MODELS



211



DISCOUNT

NUMBER



DISCOUNT

QUANTITY



DISCOUNT

(%)



DISCOUNT

COST ($)



1



0 to 999



0



5.00



2



1,000 to 1,999



4



4.80



3



2,000 and over



5



4.75



When quantity discounts are available, the purchase cost or material cost becomes a relevant cost, as it changes based on the order quantity. The total relevant costs are as follows:

Total cost = Material cost + Ordering cost + Carrying cost

Q

D

Co + Ch

Total cost = DC +

Q

2



(6-14)



where

D

Co

C

Ch



=

=

=

=



annual demand in units

ordering cost of each order

cost per unit

holding or carrying cost per unit per year



Since holding cost per unit per year is based on the cost of the items, it is convenient to express

this as

Ch = IC

where

I = holding cost as a percentage of the unit cost 1C2



The overall objective of the

quantity discount model is to

minimize total inventory costs,

which now include actual

material costs.



For a specific purchase cost (C), given the assumptions we have made, ordering the EOQ will

minimize total inventory costs. However, in the discount situation, this quantity may not be large

enough to qualify for the discount, so we must also consider ordering this minimum quantity for

the discount. A typical quantity discount schedule is shown in Table 6.3.

As can be seen in the table, the normal cost for the item is $5. When 1,000 to 1,999 units

are ordered at one time, the cost per unit drops to $4.80, and when the quantity ordered at one

time is 2,000 units or more, the cost is $4.75 per unit. As always, management must decide

when and how much to order. But with quantity discounts, how does the manager make these

decisions?

As with other inventory models discussed so far, the overall objective will be to minimize

the total cost. Because the unit cost for the third discount in Table 6.3 is lowest, you might be

tempted to order 2,000 units or more to take advantage of the lower material cost. Placing an

order for that quantity with the greatest discount cost, however, might not minimize the total

inventory cost. As the discount quantity goes up, the material cost goes down, but the carrying

cost increases because the orders are large. Thus, the major trade-off when considering quantity

discounts is between the reduced material cost and the increased carrying cost.

Figure 6.6 provides a graphical representation of the total cost for this situation. Notice the

cost curve drops considerably when the order quantity reaches the minimum for each discount.

With the specific costs in this example, we see that the EOQ for the second price category

11,000 … Q … 1,9992 is less than 1,000 units. Although the total cost for this EOQ is less than

the total cost for the EOQ with the cost in category 1, the EOQ is not large enough to obtain this

discount. Therefore, the lowest possible total cost for this discount price occurs at the minimum

quantity required to obtain the discount 1Q = 1,0002. The process for determining the minimum cost quantity in this situation is summarized in the following box.



212



CHAPTER 6 • INVENTORY CONTROL MODELS



FIGURE 6.6

Total Cost Curve for the

Quantity Discount Model



Total

Cost

$



TC Curve for Discount 3

TC Curve for

Discount 1



TC Curve for Discount 2



EOQ for Discount 2



0



1,000



2,000

Order Quantity



Quantity Discount Model

2DCo

.

B IC

2. If EOQ < Minimum for discount, adjust the quantity to Q = Minimum for discount.

Q

D

3. For each EOQ or adjusted Q, compute Total cost = DC + Co + Ch.

Q

2

4. Choose the lowest-cost quantity.

1. For each discount price (C), compute EOQ =



Brass Department Store Example

Let’s see how this procedure can be applied by showing an example. Brass Department Store

stocks toy race cars. Recently, the store was given a quantity discount schedule for the cars; this

quantity discount schedule is shown in Table 6.3. Thus, the normal cost for the toy race cars is $5.



IN ACTION



L



Lucent Technologies Develops Inventory

Requirements Planning System



ucent Technologies has developed an inventory requirements

planning (IRP) system to determine the amount of safety stock

(buffer stock) to carry for a variety of products. Instead of looking

only at the variability of demand during the lead time, the company looked at both the supply and demand of the products during the lead time. The focus was on the deviations between the

forecast demand and the actual supply. This system was used

both for products with independent demand and for products

with dependent demand.

A modified ABC classification system was used to determine

which items received the most careful attention. Items were considered both for the dollar volume and the criticality. In addition



to the Class A, B, and C categories, a D category was created to

include items that were both low dollar volume and low criticality. A simple two-bin system was used for these items.

In order to gain acceptance of the IRP system, business managers from all functions were involved in the process, and the system was made transparent so that everyone understood the

system. Because of the IRP system, overall inventory was reduced

by $55 million, and the service level was increased by 30%. The

success of the IRP system helped Lucent receive the Malcolm

Baldrige Award in 1992.

Source: Based on Alex Bangash, et al. “Inventory Requirements Planning at

Lucent Technologies,” Interfaces 34, 5 (September–October 2004): 342–352.



6.8



TABLE 6.4



USE OF SAFETY STOCK



213



Total Cost Computations for Brass Department Store



ORDER

QUANTITY (Q)



ANNUAL

MATERIAL

COST ($) = DC



ANNUAL

ORDERING

D

COST ($) = Co

Q



ANNUAL

CARRYING

Q

COST ($) = Ch

2



TOTAL ($)



DISCOUNT

NUMBER



UNIT

PRICE (C)



1



$5.00



700



25,000



350.00



350.00



25,700.00



2



4.80



1,000



24,000



245.00



480.00



24,725.00



3



4.75



2,000



23,750



122.50



950.00



24,822.50



For orders between 1,000 and 1,999 units, the unit cost is $4.80, and for orders of 2,000 or more

units, the unit cost is $4.75. Furthermore, the ordering cost is $49 per order, the annual demand

is 5,000 race cars, and the inventory carrying charge as a percentage of cost, I, is 20% or 0.2.

What order quantity will minimize the total inventory cost?

The first step is to compute EOQ for every discount in Table 6.3. This is done as follows:

EOQ 1 =

EOQ values are computed.



EOQ 2 =

EOQ 3 =



EOQ values are adjusted.



12215,00021492



B 10.2215.002



= 700 cars per order



B 10.2214.802



= 714 cars per order



B 10.2214.752



= 718 cars per order



12215,00021492

12215,00021492



The second step is to adjust those quantities that are below the allowable discount range. Since

EOQ1 is between 0 and 999, it does not have to be adjusted. EOQ2 is below the allowable range

of 1,000 to 1,999, and therefore, it must be adjusted to 1,000 units. The same is true for EOQ3;

it must be adjusted to 2,000 units. After this step, the following order quantities must be tested

in the total cost equation:

Q1 = 700

Q2 = 1,000

Q3 = 2,000



The total cost is computed.



Q* is selected.



The third step is to use Equation 6-14 and compute a total cost for each of the order quantities.

This is accomplished with the aid of Table 6.4.

The fourth step is to select that order quantity with the lowest total cost. Looking at Table 6.4,

you can see that an order quantity of 1,000 toy race cars minimizes the total cost. It should be recognized, however, that the total cost for ordering 2,000 cars is only slightly greater than the total

cost for ordering 1,000 cars. Thus, if the third discount cost is lowered to $4.65, for example, this

order quantity might be the one that minimizes the total inventory cost.

USING EXCEL QM FOR QUANTITY DISCOUNT PROBLEMS As seen in the previous analysis, the



quantity discount model is more complex than the inventory models discussed so far in this

chapter. Fortunately, we can use the computer to simplify the calculations. Program 6.3A shows

the Excel formulas and input data needed for Excel QM for the Brass Department Store problem. Program 6.3B provides the solution to this problem, including adjusted order quantities and

total costs for each price break.



6.8



Use of Safety Stock



Safety stock helps in avoiding

stockouts. It is extra stock kept

on hand.



When the EOQ assumptions are met, it is possible to schedule orders to arrive so that stockouts

are completely avoided. However, if the demand or the lead time is uncertain, the exact demand

during the lead time (which is the ROP in the EOQ situation) will not be known with certainty.

Therefore, to prevent stockouts, it is necessary to carry additional inventory called safety stock.



214



CHAPTER 6 • INVENTORY CONTROL MODELS



PROGRAM 6.3A



Excel QM’s Formulas and the Input Data for the Brass Department Store

Quantity Discount Problem

Enter demand rate,

setup cost, and

holding cost.



Enter the quantity discount

schedule of quantities and unit

prices for each price break.

Compute the order quantities for

each price break and adjust

them upward if necessary.

Compute holding,

setup, and unit cost

for each price break.



Determine the optimal order

quantity by finding the order

quantity that minimizes total

costs.



Compute the total cost

for each price break.



PROGRAM 6.3B

Excel QM’s Solution to

the Brass Department

Store Problem



When demand is unusually high during the lead time, you dip into the safety stock instead

of encountering a stockout. Thus, the main purpose of safety stock is to avoid stockouts when

the demand is higher than expected. Its use is shown in Figure 6.7. Note that although stockouts

can often be avoided by using safety stock, there is still a chance that they may occur. The

demand may be so high that all the safety stock is used up, and thus there is still a stockout.

One of the best ways to implement a safety stock policy is to adjust the reorder point. In the

EOQ situation where the demand and lead time are constant, the reorder point is simply the



6.8



FIGURE 6.7

Use of Safety Stock



USE OF SAFETY STOCK



215



Inventory

on

Hand



Time

Stockout

Inventory

on

Hand



Safety

Stock, SS



Stockout Is Avoided



0 Units

Time



amount of inventory that would be used during the lead time (i.e., the daily demand times the

lead time in days). This is assumed to be known with certainty, so there is no need to place an

order when the inventory position is more than this. However, when the daily demand or the lead

time fluctuate and are uncertain, the exact amount of inventory that will be used during the

lead time is uncertain. The average inventory usage during the lead time should be computed

and some safety stock should be added to this to avoid stockouts. The reorder point becomes

ROP = 1Average demand during lead time2 + 1Safety stock2

ROP = 1Average demand during lead time2 + SS



(6-15)



where

Safety stock is included in the

ROP.



SS = safety stock

How to determine the correct amount of safety stock is the only remaining question. Two important factors in this decision are the stockout cost and the holding cost. The stockout cost usually involves lost sales and lost goodwill, which results in loss of future sales. If holding cost is

low while stockout cost is high, a large amount of safety stock should be carried to avoid stockouts as it costs little to carry this, while stockouts are expensive. On the other hand, if stockout

cost is low but holding cost is high, a lower amount of safety stock would be preferred, as having a stockout would cost very little, but too much safety stock will result in much higher annual

holding costs.



216



CHAPTER 6 • INVENTORY CONTROL MODELS



How is the optimum stock level determined? If demand fluctuates while lead time is constant, and if both the stockout cost per unit and the holding cost per unit are known, the use of a

payoff/cost table might be considered. With only a small number of possible demand values during the lead time, a cost table could be constructed in which the different possible demand levels

would be the states of nature, and the different amounts of safety stock as the alternatives. Using

the techniques discussed in Chapter 3, the expected cost could be calculated for each safety

stock level, and the minimum cost solution could be found.

However, a more general approach is to determine what service level is desired and then to

find the safety stock level that would accomplish this. A prudent manager will look at the holding cost and the stockout cost to help determine an appropriate service level. A service level

indicates what percentage of the time customer demand is met. In other words, the service level

is the percentage of time that stockouts are avoided. Thus,

Service level = 1 - Probability of a stockout

or

Probability of a stockout = 1 - Service level

Once the desired service level is established, the amount of safety stock to carry can be found

using the probability distribution of demand during the lead time.

SAFETY STOCK WITH THE NORMAL DISTRIBUTION Equation 6-15 provides the general formula



for determining the reorder point. When demand during the lead time is normally distributed,

the reorder point becomes

ROP = 1Average demand during lead time2 + ZsdLT



(6-16)



where

Z = number of standard deviations for a given service level

␴dLT = standard deviation of demand during the lead time

Thus, the amount of safety stock is simply ZsdLT. The following example looks at how to determine the appropriate safety stock level when demand during the lead time is normally distributed and the mean and standard deviation are known.

HINSDALE COMPANY EXAMPLE The Hinsdale Company carries a variety of electronic inventory



items, and these are typically identified by SKU. One particular item, SKU A3378, has a demand that is normally distributed during the lead time, with a mean of 350 units and a standard

deviation of 10. Hinsdale wants to follow a policy that results in stockouts occurring only 5% of

the time on any order. How much safety stock should be maintained and what is the reorder

point? Figure 6.8 helps visualize this example.

FIGURE 6.8

Safety Stock and the

Normal Distribution

5% Area of Normal Curve

SS

μ = 350



X =?



μ = Mean demand = 350

σ = Standard deviation = 10

X = Mean demand + Safety stock

SS = Safety stock = X – μ = Zσ

Z =



X–μ

σ



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