"Inventory Management" 
Index to Module Five Notes 
"When the tires get down to 4, it's time to order a whole lot more."From a sign posted in Lieutenant Harrington's
Headquarters' Hooch somewhere in the Central
Highlands, South Vietnam, 1967
5.1: Economic Order Quantity
(EOQ) Model
Introduction
Since 1915, management scientists have
been applying quantitative methods to help inventory managers make
two critical decisions: how much inventory to order, and when to
order it. With low value items, the how much to order and when
decisions can be based on simple heuristics or rules of thumb. While
the opening quoted order quantity decision rule was meant as my
attempt at humor in the headquarters, it does serve the purpose of
illustrating a simple rule of thumb for $25.00 parts.
Twentyfive years later I found myself planning inventory management
strategy for Johnson and Johnson Medical as part of a University of
South Florida research grant. This was quite a different situation.
From $25.00 tires I went to $25,000 components. If the annual cost to
carry inventory is 20% of the item value, one can quickly see that
more advanced decision rules are needed to make inventory management
decisions to meet demand while minimizing inventory costs.
Yet, when I went into the warehouse, it did not seem like much
attention was being given to effective inventory management. There
was inventory all over the place. I asked why? The answer was typical
of US manufacturers in the 1980's, "we have this extra inventory
"just in case" the vendor makes a late delivery." The
purchaser went on to explain that the average cycle time to get
components into the Oldsmar assembly facility was 21 days (cycle time
is the time between when items are ordered until they are received).
However, the standard deviation was 3 days, meaning they could
experience cycle times as long as 30 days (average plus 3 standard
deviations).
The Oldsmar operation involved assembly of components into sterilized
surgical kits which were then sold to hospitals around the world.
Since the assembly operation was on a tight schedule based on the 21
day inbound cycle times, shipments 9 days late were unacceptable.
So.... you guessed it, the assembly plant carried 9 extra days worth
of inventory "just in case" the shipment was late. That 9 extra days
of inventory was costing them $125,000 every quarter in carrying
cost.
These are real costs  if the company did not have those 9 days of
inventory on the shelf, they could do alternative things with the
money.
The Japanese taught us that rather than have "just in case there is a
problem" inventory, solve the problem and switch to "just in
time" management strategy. Not all cultures or systems need
to be operated "just in time" but cycle time compression has been the
focus of a lot of business strategies this past decade.
Well, there was a happy ending. We gave some incentives to the
vendors (anything to save that $125,000 carrying cost), put in some
quantitative methods to help make inventory management decisions, and
rationalized the vendor base. The net result was long term savings in
inventory costs.
Inventory is one luxury that product businesses have over service
businesses. As we discussed in the last module, services cannot be
inventoried, whereas raw material, components, work in process, and
finished goods can be buffered anywhere up and down the supply chain.
Sometimes the buffer is bad, such as to guard against problems in
delivery, forecasting, and what ever. Sometimes the buffer is good,
such as cycle stock or the stock needed to maintain operations; or
product line inventory in the retail store to attract customers. How
much of that buffer to order and when is the subject of this
module.
Inventory Management Considerations
Inventory consists of a stock of items, the size of which is
called the inventory level. Inventory management
involves making decisions concerning how much inventory
to order and when. The basic criterion in making these
decisions is to minimize total inventory costs, such as the cost to
carry inventory, the cost to order inventory, and the item cost,
subject to meeting demand for the items. Inventory
control involves process, procedures, and infrastructure to
maintain the inventory at the desired level. Inventory management is
the subject of this module. You will visit some inventory control
considerations in the operations management course.
The quantitative methods designed to help make effective inventory
management decisions apply to independent demand items.
A simple example of an independent demand item is the automobile.
Demand for the automobile is forecasted based on past
demand, market conditions, and so forth. On the other hand, demand
for the two axles for the automobile is derived from
the demand for the automobile. Thus the demand for axles is
dependent demand. Material requirements
planning accounting models apply to management of dependent demand
items and are not covered in this module. Those methods are covered,
however, in operations management courses that are offered in many
MBA programs (including ours).
When we speak of inventory, we generally refer to finished goods, raw
material, spare purchased parts and supplies. But we could also speak
of blood in a blood bank, cash on hand at a bank, and so forth.
The general purposes for carrying inventory may include one or more
of the following. Inventory may:
1. Forming the basis for doing business. This might be the product line inventory in the retail store.
2. Provide a favorable return on investment. Sometimes inventories have value and are traded for that value.
3. Allow the buyer to take advantage of quantity discounts. It may be cost effective to buy in quantities larger than the cycle stock and store the excess because the discount for buying in large lots is substantial.
4. Protect against fluctuations in demand, delayed supply, and inflation.
5. Buffer operations from the customer. For example, a manufacturer may wish to make a long production run for economies of scale in production. The production run may result in more items than current demand. The excess is placed in inventory until the next demand period.
Inventory Costs
For the basic economic order quantity model, there are two
variable costs to be minimized: the cost to hold
inventory and the cost to order inventory. Generally,
the unit of time measurement is annual.
Total Inventory Costs = Holding Cost + Ordering Cost
The cost relationships can be sketched out in a graph with dollars on the vertical axis and the Order Quantity on the horizontal axis, as shown in Figure 5.1.1.
Figure 5.1.1
Dollars


/
Holding Cost


/


/

\
/

\
/

\
/

+ 
/
\_
__
__
__
__
__
Ordering Cost

/

/
_
__
__
__
__
__
__
__
__
__
__
Order Quantity
This chart illustrates that as the order
quantity (number of items in an order) increases, the cost to order
decreases and the cost to hold inventory increases on an annual
basis. Note how these cost curves model the same behavior as the
waiting and service cost curves of the waiting line systems. Both
systems reflect points at which total costs can be minimized  where
we strike a balance between the two cost curves. More on this
later.
Inventory Pattern
Inventory is depleted as demand occurs. For example, let's assume
that annual demand (D) for an item in inventory is
55,000. The firm could order all 55,000 items at the beginning of the
year and hold them in inventory to meet demand throughout the year.
Assuming that the firm works 250 days a year (unlike university
students and faculty, the firm takes weekends and holidays off), the
daily demand (d) is:
d = D / days in year = 55,000 / 250 = 220 items per day
Side note: I just placed my first "dot com"
book order over the Internet  I placed it on a Sunday afternoon, and
the order was processed and shipped that day  there goes weekends
and holidays in the "dot com" world.
Back to the course... We can present a picture of this in a graph,
with the vertical axis being inventory quantity, and the horizontal
axis being time. The plan is that inventory just runs out as we reach
day 250, and the next year's order arrives that day. The order is for
55,000 (called Order Quantity, Q) which is depleted at
the rate of 220 per day. Figure 5.1.2 also shows the average
inventory on hand as one half of the 55,000 or 27,500. We will see
later that this average is important as it is the basis for computing
inventory carrying cost.
Figure 5.1.2
Quantity   55,000 \ \  \  \  \   \  Avg. Q     \      \   \   \  _ __ __ __ __ __ __ __ \_ __ __ 250 Time
Because it may be expensive to hold large quantities of inventory, the alternative is to order more often, thus holding less inventory in each order. Suppose the firm decides to order 5 times (number of orders, N = 5), then the order quantity, Q, is:
Q = D / N = 55,000 / 5 = 11,000
The daily demand rate would 11,000 / 250 or 44 items per day. Ordering five times a year, would give us an order cycle time (t) of:
t = number of days in year / N = 250 / 5 = 50 days
The order cycle time is the time between when an order arrives until it is depleted. Thus, orders need to be scheduled to arrive at day 50, 100, 150, 200 and 250 in order to replenish the inventory as soon as the orders are depleted. The inventory pattern for this order quantity strategy is shown in Figure 5.1.3.
Figure 5.1.3
Quantity    11,000 \ \ \ \ \ \  \  \  \  \  \  \  \  \  \  \  \  _ __ __ \_ __ __ \_ __ __ \_ __ __ \_ __ __ \_ __ 50 100 150 200 250 Time
This strategy would result in an average
inventory of 1/2 Q or 5,500. The holding costs would be less than
ordering once a year, but the ordering costs would be five times as
much. That is the essence of the problem: how to balance the cost to
hold inventory against the cost to order inventory. Let's examine
this balance in the classic Economic Order Quantity Model (EOQ).
Economic Order Quantity Model: How Much to Order
In order for inventory managers to make use of the classic EOQ
Model to determine how much inventory to order and when, they must be
willing to accept the assumptions. These assumptions may seem
restrictive but two points are important to consider. First, total
inventory costs are fairly insensitive to deviations from the optimal
order quantity as long as the deviations are not too drastic. Thus,
while an assumption may not be precisely met, the impact may not be
too great. Second, there are variations to the classic EOQ Model that
allow us to relax the assumptions. I make a note when this is
possible in the following brief discussion of each
assumption.
Assumptions:
1. The daily demand rate, d, is constant and independent. We will examine a way to work around this assumption when we examine an extension to the EOQ model to consider variation in the demand rate.
2. The order quantity, Q, is constant for each order and the entire order is received at one time. We will look at another model (Economic Production LotSize Model) that allows the order to be received over time. We will also discuss a model that allows for varying of the order quantity, as is typically done with relatively inexpensive items.
3. The cost per order, C_{o}, is constant and does not depend on the size of the order.
4. The unit cost, C, of the inventory item is constant and does not depend on the size of the order. We will examine a quantity discount model that allows adjustments in the unit costs as more units are ordered each time an order is placed.
5. The inventory holding cost per unit per time period, C_{h}, is constant.
6. Shortages such as backorders and stock outs are not permitted. We will examine a model that allows for backorders.
7. The lead time for an order is constant. Lead time is the time between when an order is placed until it is received. We will consider an extension to the EOQ Model that allows for variability in lead time.
8. The inventory level is reviewed on a continuous basis. This is generally the case when we are considering relatively expensive items. When the items are relatively inexpensive, the inventory manager may often review the inventory level every fixed time period, like once a month, and then place an order based on that level.
9. The planning horizon consists of multiple timeperiods. This simply means that the item in inventory has shelf life and may be inventoried for more than one period in time. Thus, the EOQ Model does not work for daily newspapers or vegetables or any other item that has a one period shelf life.
The inputs that an inventory manager must consider in using the EOQ
Model to make the order quantity decision include the annual demand
(D), number of days in the year, order lead time (m), unit cost (C),
holding cost rate (I), and the order cost (C_{o}). We have
discussed the annual demand, lead time, unit cost, and days in the
year components.
The holding cost rate is used to compute the holding cost
component, C_{h}. Holding costs may include the cost
of financing the inventory investment. This cost of
capital represents the cost associated with capital tied up
in inventory. The cost of capital is usually expressed as a
percentage of the amount invested, such as 20%. Other, usually
smaller holding costs may include insurance, taxes, pilferage and
warehouse overhead  costs that vary with the inventory investment.
Thus, the holding cost rate (I) may total 20% to 25% of
the inventory investment, depending upon a firm's particular
situation.
The order cost, C_{o}, covers preparation
expenses for the order, such as payment, communication, invoice
verification, receiving and so on. Technology has had a major impact
on this expense when you compare green eyeshade days of manual order
processing 30 years ago, to ecommerce and its electronic data
interchange of today.
Here are the inputs for an example problem:
Annual Demand = D = 3,200 units
Working Days = 250
Unit Cost = C = $18
Holding Cost Rate = I = 22% per dollar of inventory investment peryearOrder Cost = C_{o} = $75 per order
Other inputs needed for the EOQ Model are derived from these inputs:
Daily Demand = d = D / Working Days = 3,200 / 250 = 12.8
Holding Cost = C_{h} = I C = .22 * $18 = $ 3.96
The main output variable is the order quantity, Q, how much to order every time an order is placed. This can be determined by the EOQ Model or set by the inventory manager/customer. Suppose the inventory manager decides to place orders twice a month, or 24 times a year. From that decision, we can determine the order quantity, Q:
Order Quantity = Q = D / N = 3,200 / 24 = 133 (note: I rounded 133.33 to 133)
Once we know Q, we can determine the total inventory costs (TIC) for the order strategy of placing 24 orders, each with an order quantity of 133:
TIC = Annual Holding Cost + Annual Ordering Cost
TIC = (0.5) ( Q )(C_{h}) + (D / Q) C_{o}
TIC = (0.5)(133)(3.96) + (3,200 / 133) 75
TIC = 263 + 1,805 = $2,068
Total costs are the criteria by which inventory managers measure the results of a particular order quantity strategy. With this strategy, the inventory manager notices that the ordering cost exceed the holding cost and makes an adjustment to order less times and carry more inventory with each order. Suppose the decision is to order once a month, or 12 times a year.
Order Quantity = Q = D / N = 3,200 / 12 = 267
Once we know Q, we can determine the total inventory costs:
TIC = Annual Holding Cost + Annual Ordering Cost
TIC = (0.5) ( Q )(C_{h}) + (D / Q) C_{o}
TIC = (0.5)(267)(3.96) + (3,200 / 267) 75
TIC = 529 + 899 = $1,428
Table 5.1.1 illustrates three other strategies,
ordering one time a year, ordering 50 times a year (every week minus
the two weeks the firm shuts down), and ordering daily (sort of like
a "just in time" strategy).
Table 5.1.1
of Orders 

(0.5)(Q)(C_{h}) 
(D/Q) C_{o} 
Inventory Cost 

























This table is instructive. It begins to trace
out the relationship between holding cost and ordering cost, which
take the form shown in Figure 5.1.1. Note how holding cost increases
as Q increases, and how order cost decreases as Q increases. The
minimum total cost appears at an order quantity of 267, representing
12 orders. Also note that at this minimum total cost, the holding and
order costs are closer than at any other total cost.
In fact, Figure 5.1.1 tries to show that the minimum total cost
occurs where order cost equals holding cost. This important
observation is the basis for the EOQ Model. To derive the EOQ Model,
set order cost equal to holding cost and solve for the decision
variable Q. Holding cost is average inventory times the holding cost
per dollar investment in inventory per year. The average inventory
level is (beginning inventory + ending inventory) / 2 = ( Q + 0 ) / 2
or Q/2. Remember from the assumptions that beginning inventory is Q,
the order quantity, and ending inventory is 0 right before the next
order quantity arrives. Order cost is the cost per order times the
number of orders (D / Q).
Holding Cost = Ordering Cost
(0.5) ( Q )(C_{h}) = (D / Q) C_{o}
(0.5) (Q^{2}) (C_{h}) = D C_{o }Q^{2} = ( 2 D *C_{o} ) / C_{h}
EOQ = Q = Square Root [( 2 D *C_{o} ) / C_{h} ]
This is pretty slick, huh! With the EOQ formula the inventory manager doesn't have to use the trial and error approach of Table 5.1.1 in trying to find the right Q... just use the EOQ formula. For this example problem:
EOQ = Q = Square Root [( 2 * 3,200 * 75 ) / 3.96 ] = 348
Now we can compute the total cost for this order quantity designed to minimize total inventory cost (TIC):
TIC = (0.5) ( Q )(C_{h}) + (D / Q) C_{o}
TIC = (0.5)(348)(3.96) + (3,200 / 348) 75
TIC = 690 + 689 = $1,379
Note that at the optimal order quantity,
holding cost is equal to order cost (the slight difference is due to
rounding. Also note, that the optimal solution of a Q = 348 is not
too far from the trial and error solution of Q = 267. In other words,
we had a 30% increase in the order quantity, but only a 5% change in
costs. This confirms my point earlier that with shallow dishshaped
total cost functions, decision variable deviations result in minor
changes to total cost as long as we keep in the area of optimal Q.
This is important information for the inventory manager since they
may not always be able to order 348 units. What if the vendor only
sells packages of 100 (100, 200, 300, etc.). The inventory manager
knows that he or she could order 300 instead of 348 and not deviate
too far from optimal costs.
Once we know the optimal order quantity, we can derive several other
inventory management results:
Number of Orders = N = D / Q = 3,200 / 348 = 9.2 orders per year
Cycle time = t = Working Days / N = 250 / 9.2 = 27
Maximum Inventory Level = Q = 348
Average Inventory Level = Q/2 = 348 / 2 = 174
Even though the EOQ Model is based on fairly
straightforward mathematical derivations, inventory managers rely on
software to do the computations.
Using The Management Scientist Software Package
To use The Management Scientist Inventory Module to solve
for EOQ, the related output results, and inventory costs, click
Windows Start/Programs/The Management Scientist/The Management
Scientist Icon/Continue/Select Module 8 Inventory/OK/File/New and
you are ready to load this example problem.
The next dialog screen asks you to select a model. Highlight
"Economic Order Quantity." Next comes the input dialog screen.
Enter 3,200 for Demand, 75 for Ordering Cost, keep
percent for Holding Cost selection and enter 22, enter 18 as
the Unit Cost, select Compute Reorder Point, and enter
5 for the lead time. Then select Solve, and you should get the
following:
Printout 5.1.1
INVENTORY MODEL
***************
ECONOMIC ORDER QUANTITY
***********************
YOU HAVE INPUT THE FOLLOWING DATA:
**********************************
ANNUAL DEMAND = 3200 UNITS PER YEAR
ORDERING COST = $75 PER ORDER
INVENTORY HOLDING COST:
A. ANNUAL INVENTORY CARRYING CHARGE = 22.0%
B. COST PER UNIT = $ 18 PER UNIT
WORKING DAYS PER YEAR = 250 DAYS
LEAD TIME FOR A NEW ORDER = 5 DAYS
INVENTORY POLICY
****************
OPTIMAL ORDER QUANTITY 348.16
ANNUAL INVENTORY HOLDING COST $689.35
ANNUAL ORDERING COST $689.35
TOTAL ANNUAL COST $1,378.70
MAXIMUM INVENTORY LEVEL 348.16
AVERAGE INVENTORY LEVEL 174.08
REORDER POINT 64.00
NUMBER OF ORDERS PER YEAR 9.19
CYCLE TIME (DAYS) 27.20
We computed these same output results with
the exception of the Reorder Point, which is covered in the next
subsection.
With The Management Scientist Software, the inventory manager
can quickly conduct the always important sensitivity analysis. For
example, what is the impact on Q and total costs of changing the
holding cost percent to 28%, what is the impact on Q and total cost
of a 25% decrease in demand, and so forth.
Pause and Reflect
In order to minimize total inventory costs, management should plan on ordering 348 units 9 times a year. This gives an average inventory of 174 units, a cycle time, or time between orders of 27 working days, and total inventory costs of $1,379. The assumptions include constant demand, fixed order quantity, order receipt at one time, are that demand, constant order and holding costs, constant unit price, no shortages, continuous review of the inventory and multiple period planning horizon. One other assumption concerns the lead time which is discussed in the next subsection.
When to Order
When to place an order in a continuous review inventory system is
a function of the lead time, or how much time it takes between
ordering the units and receiving the units, and the daily demand
rate. The inventory manager wants to make sure that after the order
is placed, there is enough inventory to cover the daily depletion
until the order arrives.
For this problem, the lead time (m) is 5 days. Earlier, we computed
the daily demand rate to be 12.8. The reorder point, in terms of the
inventory level, is the daily demand rate times the lead
time:
Reorder Point = r = dm = 12.8 * 5 = 64
That's it for the EOQ Model and the inventory
management decisions that must be made under the strictest set of
assumptions. The inventory manager now has answers to the two
critical questions: how much to order (348 units), and when (when the
inventory level reaches 64 units. While it is important to know the
lead time in days for processing purposes, the lead time has to be
expressed in units for continuous review purposes. Order Lead Daily Lead Time 1 5 12.8 64 2 5 8.8 44 3 4 21 84 4 5 12.8 64 5 5 10.4 52 6 6 12.7 76 7 5 12.8 64 8 5 11.6 58 9 4 17.5 70 10 5 12.8 64 11 5 9.2 46 12 4 20.5 82 13 5 12.8 64 14 5 11.6 58 15 6 11.7 70 16 5 12.8 64 17 5 10.4 52 18 5 15.2 76 19 4 16 64 20 6 10.7 64
Did you ever visit a hardware store and after you picked an item off
of a display hanger you noticed a tag stating "reorder point"  that
is what we are talking about. If you haven't done that  may be you
would enjoy a visit to a local hardware store and see for
yourself!
Reorder Point Under Conditions of Uncertainty
A more realistic situation may be that either the lead time, or
the daily demand, or both exhibit variability which must be accounted
for in the reorder point formula. This section addresses this issue
which concerns relaxing the first assumption (constant daily demand
rate) and seventh assumption (constant lead time) introduced earlier
in this section. Accounting for variability in the formula is easy,
data gathering to determine the variability is the hard part.
Suppose the firm experienced the following actual lead times and
daily demands after the reorder point was reached for the last 20
order cycles.
Table 5.1.2
Cycle
Time
Demand
Demand
Note again that lead time demand is a function
of both the lead time and the daily demand (lead time demand = lead
time times daily demand). The lead time stays fairly constant, as it
should, or the firm should shop around for a new vendor or give some
incentive to the vendor to get the lead time fixed. This might entail
going to more dedicated forms of transportation. The daily demand
variability is harder to control in the short term. In the long term,
demand can be changed by changing the price, the quality and so
forth.
The variable of interest is the lead time demand since that effects
the order point. The average lead time demand can be computed as 64
(Mean lead time demand = (64 + 44 + 84 + 64 + ... + 64) / 20). Note
earlier that the reorder point under conditions of certainty was 64
(lead time times the daily demand). Also note that during some order
cycles, the lead time demand was greater than 64.
For example, in order cycle number 18 the lead time was right on
target at five days, but the daily demand came out to be 15.2 days.
This gives a mean time demand of 76 units  fully 12 more units above
the expected lead time demand of 64. If demand is greater than
expected after an order is placed, or the lead time is longer than
expected after an order is placed, or the lead time demand is greater
than the order point lead time demand  guess what... that's right,
the firm experiences a stock out. Of course, the opposite is true.
If, after an order is placed, the demand slows down and/or the lead
time is shorter than expected, then more inventory than expected will
be on hand when the order arrives. Generally, being out of stock is
more critical than having some level of additional stock.
To guard against a stock out under conditions of uncertainty, firms
add safety stock to the mean lead time demand. Here is the
formula:
Reorder Point = Mean Lead_{ Time Demand} + Safety Stock
Reorder Point = 64 + (Z Score times Std Dev_{Lead Time Demand})
From Table 5.1.2, we can compute the standard deviation of lead time demand to be 10.7. The Z score is how we translate the customer service objective into the formula. Suppose the firm wants to provide a customer service level of 97.5 percent. This means, that the firm wants to be able to meet 97.5 percent of the demand. This could be 97.5 percent of the demand during one order cycle, or 97.5 of the order cycles will have demand met, or however the firm wants to operationalize the customer service level. Another way of looking at this is to express the 97.5 percent customer service level as the complement: there is a 2.5 percent chance of a stock out. In Excel, if you enter =NORMSINV(0.975) in an active cell, you get a Z Score of 1.96 (almost 2). The reorder point is:
Reorder Point = 64 + (1.96 * 10.7) = 64 + 21 = 85
Do you see what we have done? We set the
reorder point higher than the expected lead time demand to guard
against stock outs. The safety stock of 21 will be used when the lead
time is shorter than expected, and/or the daily demand is greater
than expected. The safety stock will be replenished when the lead
time is shorter than expected, and/or daily demand is less than
expected.
But there is "no free lunch." The safety stock incurs inventory
holding costs:
Safety Stock Holding Costs = C_{h} * Safety Stock
Safety Stock Holding Costs = $3.96 * 21 = $83.16.
This may not seem much to guard against a stock out  but what if the firm managed 2,000 stock keeping units  those safety stock costs add up. In my experience in the US Air Force, we only went to 95% or 97.% customer service levels in wartime. During peacetime, we ran 85% customer service levels. For this firm, an 85% customer service level gives the following reorder point and costs:
Reorder Point = 64 + (1.04 * 10.7) = 64 + 11 = 75
Safety Stock Holding Costs = $3.96 * 11 = $43.56
Setting safety stock is just like adding a
buffer to a forecast. Whether a firm uses a formal mathematical
approach or judgment, we often find safety or buffer stocks to guard
against stock outs in inventory systems.
The Management Scientist has an inventory module that allows
us to incorporate
probabilistic demand during lead time into the EOQ
orderquantity/reorder point model. This is illustrated for the above
example, using the 97.5% customer service level (2.5% chance of a
stock out) in Printout 5.1.3 below.
Printout 5.1.3
INVENTORY MODEL
***************
ORDER QUANTITYREORDER POINT WITH PROBABILISTIC DEMAND
******************************************************
YOU HAVE INPUT THE FOLLOWING DATA:
**********************************
ANNUAL DEMAND = 3200 UNITS PER YEAR
ORDERING COST = $75 PER ORDER
INVENTORY HOLDING COST:
A. ANNUAL INVENTORY CARRYING CHARGE = 22.0%
B. COST PER UNIT = $ 18 PER UNIT
WORKING DAYS PER YEAR = 250 DAYS
LEAD TIME DEMAND DISTRIBUTION IS NORMAL WITH
A. MEAN = 64 UNITS
B. STANDARD DEVIATION = 10.7 UNITS
PROBABILITY OF A STOCKOUT = 0.025
INVENTORY POLICY
****************
OPTIMAL ORDER QUANTITY 348.16
ANNUAL INVENTORY HOLDING COST $772.82
ANNUAL ORDERING COST $689.35
TOTAL ANNUAL COST $1,462.17
MAXIMUM INVENTORY LEVEL 369.23
AVERAGE INVENTORY LEVEL 195.16
REORDER POINT 85.08
NUMBER OF ORDERS PER YEAR 9.19
CYCLE TIME (DAYS) 27.20
SAFETY STOCK 21.08
ANNUAL SAFETY STOCK COST $83.47
EXPECTED STOCKOUTS PER YEAR .23
PROBABILITY OF A STOCKOUT PER CYCLE .0250
Discount Cost (C)
Quantity Discounts
The next assumption that we want to relax is assumption number four 
constant unit cost, C. The idea here is that the vendor may give a
price discount to the firm if the firm would agree to buy larger
order quantities. This is called the "supply chain shell game" or who
can we get to hold the inventory in the supply chain. When a vendor
encourages a firm to order in larger quantities, the vendor is simply
shifting the holding costs to the next node in the supply chain.
Suppose the vendor offers the firm the following discount
schedule:
Table 5.1.3
Recall that we computed the EOQ as 348 for the unit cost of $18. The
total inventory costs for this EOQ was:
TIC = Annual Holding Cost + Annual Ordering Cost
TIC = (0.5) ( Q )(C_{h}) + (D / Q) C_{o}
TIC = (0.5) ( Q )(I * C) + (D / Q) C_{o}
TIC = (0.5) (348) (0.22 * 18) + (3,200 / 348 ) 75
TIC = $1,379
Note that I substituted the holding cost interest rate times the unit cost for C_{h} so that we can prepare for analyses in which the unit cost changes. The total cost for this inventory order quantity, including demand times unit cost is:
TC = TIC + ( D * C ) = $1,379 + (3,200 * 18) = $1,689 + $57.600TC = $58,979
Does the firm want to take a discount? To answer this question, we run the new numbers at the discount rates. Let's look at the 5% discount first, which gives a new unit cost of $17.1. The first step is to compute the EOQ.
EOQ = Square Root [ ( 2 D * C_{o} ) / ( I * C) ]
EOQ = Square Root [ ( 2 * 3,200 * 75 ) / ( 0.22 * 17.1) ]
EOQ = 358
The EOQ comes out to be less than the minimum
order size to get the discount price so we adjust up to the
order size needed to get the 5% discount. That is, the firm
should order 1,000 units. The reason that we do not go higher than
the minimum order size for the discount is that the optimum
order quantity is 358 and we want to stay as close to the
optimum as possible while still getting the price discount.
Letting Q = 358, we now solve for total inventory and total
costs:
TIC = (0.5) ( Q )(I * C) + (D / Q) C_{o}
TIC = (0.5) (1,000) (0.22 * 17.10) + ( 3,200 / 1,000 ) * 75
TIC = $1,881 + 240
TIC = $2121
TC = TIC + (D * C) = $2121 + (3,200 * 17.10) = $56,841
Here, the firm can save by taking the 5%
discount. Even though the total inventory cost is higher because the
firm uses an order quantity above the EOQ, the discount was
significant enough to result in a total savings compared to not
taking the 5% discount. Discount Cost (C) (to get discount)
The results obtained so far are shown in Table 5.1.4. Also included
in this table is an evaluation of the EOQ, adjusted order quantity
and total cost for the 10% discount. The computations are made in a
manner similar to the evaluation of the 5% discount.
Table 5.1.4
Note how the TC curve reached a minimum at the 5% discount level,
then began to climb as the holding cost for ordering in quantities of
3,000 exceeded the price discount savings.
Fortunately, when purchasers negotiate with vendors the discount
schedule, software is available for the rapid computation of the
order quantities and costs associated with various discount
schedules. The Management Scientist Inventory Module includes
an EOQ with Quantity Discounts option. The output for this option is
shown in Printout 5.1.2 for the above example.
Printout 5.1.2
INVENTORY MODEL
***************
ECONOMIC ORDER QUANTITY WITH QUANTITY DISCOUNTS
***********************************************
YOU HAVE INPUT THE FOLLOWING DATA:
**********************************
QUANTITY DISCOUNT INFORMATION
CATEGORY UNIT COST MINIMUM QUANTITY
  
1 $18.00 0
2 $17.10 1000
3 $16.20 3000
ANNUAL DEMAND = 3200 UNITS PER YEAR
ORDERING COST = $75 PER ORDER
ANNUAL INVENTORY CARRYING CHARGE = 22
WORKING DAYS PER YEAR = 250 DAYS
INVENTORY POLICY
****************
OPTIMAL ORDER QUANTITY 1,000.00
ANNUAL INVENTORY HOLDING COST $1,881.00
ANNUAL ORDERING COST $240.00
ANNUAL PURCHASE COST $54,720.00
TOTAL ANNUAL COST $56,841.00
MAXIMUM INVENTORY LEVEL 1,000.00
AVERAGE INVENTORY LEVEL 500.00
NUMBER OF ORDERS PER YEAR 3.20
CYCLE TIME (DAYS) 78.13
The next order of business is to examine
two more classic inventory models: the production lot size model and
the planned shortages model.
Pause and Reflect
The classic EOQ model determines two inventory management strategies: how much to order when an order is placed, and when to place the order. The classic EOQ model assumes the following are constant: daily demand, order quantity, cost per order, unit cost, holding cost, and lead time. We also examined two extensions where we first relaxed the assumption that daily demand and lead time were constant which lead to adjustments in the "when to order" decision; and second where we relaxed the assumption that the unit cost was constant which lead to adjustments in the order quantities.
5.2: Economic Production Lot
Size Model
The next model was originally designed
for situations in which the inventory is produced by
the firm, rather than ordered by the firm. As such, the
order cost changes to a setup cost. Since production
runs generally take some time between equipment setup and production,
the order is received over time, rather than at one point in time. In
the classic production lot size model, this is called the
production rate.
This model can actually be used for situations in which the firm
orders their inventory, rather than produces it, and operates under
the condition that the order is not received at one point in time but
over time. This is a very common. Perhaps a partial order is received
on Monday, another partial order on Tuesday, and the third partial
order on Wednesday. Thus, the assumption pertaining to the order
received at one point in time (part of Assumption Two in Module 5.1
Notes) is relaxed in this model.
We will use The Management Scientist to examine a production
lot size model example input and output. The interested reader is
referred to pages 541543 in the text for the formulas from which the
output is derived. In Printout 5.2.1, note we are asked to provide a
setup cost rather than an order
cost. Also note that we are asked to provide an annual production
rate. The setup cost and production rate would apply to situations
where the firm produced the item. If the firm ordered the item, then
the setup cost is simply the order cost from Module 5.1 Notes, and
the production rate is the rate at which units are received over
time. The vendor would have to provide this information. With 250
working days in the year, I am assuming for this example that the
daily receipt (production) rate is 12,000 / 250 or 48 units a day.
All other input is the same as the example in the EOQ illustration of
Module 5.1 Notes.
Printout 5.2.1
INVENTORY MODEL
***************
ECONOMIC PRODUCTION LOT SIZE
****************************
YOU HAVE INPUT THE FOLLOWING DATA:
**********************************
ANNUAL DEMAND = 3200 UNITS PER YEAR
ANNUAL PRODUCTION RATE = 12000 UNITS PER YEAR
SETUP COST = $75 PER SETUP
INVENTORY HOLDING COST:
A. ANNUAL INVENTORY CARRYING CHARGE = 22.0%
B. COST PER UNIT = $ 18 PER UNIT
WORKING DAYS PER YEAR = 250 DAYS
LEAD TIME FOR A NEW ORDER = 5 DAYS
INVENTORY POLICY
****************
PRODUCTION LOT SIZE 406.56
ANNUAL INVENTORY HOLDING COST $590.32
ANNUAL SETUP COST $590.32
TOTAL ANNUAL COST $1,180.64
MAXIMUM INVENTORY LEVEL 298.14
AVERAGE INVENTORY LEVEL 149.07
REORDER POINT 64.00
NUMBER OF SETUPS PER YEAR 7.87
CYCLE TIME (DAYS) 31.76
The output shows the order quantity is now 406.56 versus 348 when
using the EOQ model. Note carefully that the maximum inventory level
and average inventory level are less than 348 and 174, respectively,
under the EOQ model. This is because the units are not received
all at once, but rather over time. While items are being received
over time, demand continues thereby resulting in a reduction in the
maximum and average inventories. With this reduction in inventory
levels, comes a reduction in holding cost.
The order (or production) cost is also lower since there are fewer
orders. The total inventory cost with the economic production lot
size model is $1,180 compared to $1,379 with the EOQ model as a
result. Of course, the firm would have to carefully evaluate using
the same holding cost and order cost when the shipment is received
over time since there may be more cost associated with monitoring and
inchecking orders over time rather than all at once.
5.3: Planned Shortages
The last model we will examine relaxes
the assumption that there can be no shortages (Assumption Number 6 in
Module Notes 5.1). This means that backorders and lost sales can
occur. While firms generally do not plan an inventory management
strategy of lost sales, many do plan a strategy of allowing
backorders. For example, when is the last time you when to Service
Merchandise, saw an item you liked on a shelf, filled out the paper
work for the item, and upon checking out were told they were out of
stock of that item  would you like a substitute item or would you
like the item delivered to your home in 4 days via UPS? Guess what,
Service Merchandise just transferred their inventory holding cost to
you. This works as long as Service Merchandise knows
that most of their customers will accept either a substitute or will
wait for 4 days  otherwise a backorder becomes a lost sale.
As with the economic production lot size model, we will examine the
input and output of the planned shortages model by using The
Management Scientist. The interested reader is referred to
pp. 544548 in the text for the formulas from which the output is
derived. Let's continue with the example problem from Module 5.1
Notes. This time, assume that backorders are allowed. All of the
input is the same as the example in Printout 5.1.1, except we added a
backorder cost of $5. This could be for the extra paper work in
handling a backorder, and/or the extra level of transportation, such
as by UPS.
Printout 5.3.1
INVENTORY MODEL
***************
ECONOMIC ORDER QUANTITY WITH PLANNED SHORTAGES
**********************************************
YOU HAVE INPUT THE FOLLOWING DATA:
**********************************
ANNUAL DEMAND = 3200 UNITS PER YEAR
ORDERING COST = $75 PER ORDER
INVENTORY HOLDING COST:
A. ANNUAL INVENTORY CARRYING CHARGE = 22.0%
B. COST PER UNIT = $ 18 PER UNIT
BACKORDER COST = $5 PER UNIT PER YEAR
WORKING DAYS PER YEAR = 250 DAYS
LEAD TIME FOR A NEW ORDER = 5 DAYS
INVENTORY POLICY
****************
OPTIMAL ORDER QUANTITY 466.06
ANNUAL INVENTORY HOLDING COST $287.36
ANNUAL ORDERING COST $514.95
ANNUAL BACKORDER COST $227.59
TOTAL ANNUAL COST $1,029.91
MAXIMUM INVENTORY LEVEL 260.08
AVERAGE INVENTORY LEVEL 72.57
MAXIMUM BACKORDERS 205.98
REORDER POINT * 141.99
NUMBER OF ORDERS PER YEAR 6.87
CYCLE TIME (DAYS) 36.41
* NOTE: THE NEGATIVE REORDER POINT INDICATES THAT THE ORDER
SHOULD BE PLACED WHEN THE NUMBER OF BACKORDERS =141.99
This model results in a total inventory
cost savings when compared to the EOQ model without planned shortages
in Printout 5.1.1. That's because the inventory is carried as
backorders rather than as units in storage so the inventory holding
cost is reduced more than the additional backorder cost. Of course,
the firm runs the risk that backorders can become lost sales, then
the $5 backorder cost is unrealistic.
That completes our coverage of the classic quantitative methods
designed to help inventory managers determine how much inventory to
order, and when to order it. These methods involve a fair amount of
computation so inventory managers generally rely on software for the
"number crunching." Because of their sophistication, these models are
generally used for relatively high cost items  those items that
warrant continuous review.
For low cost items, the inventory manager may wish to use a simple
periodic review inventory model as discussed in pages 561 564 in the
text. That is, instead of continuous review of the inventory (which
is required to know when the reorder point is reached), the inventory
manager may elect to view (count) the inventory at set time
intervals, such as once a month. Then, an order is placed for the
difference between the current inventory level and the maximum
desired inventory level.
For very inexpensive items the inventory manager may wish to use even
a similar model such as a "twobin" system. For example, in the
hardware store, penny nails may be on display in a bin. When the bin
is empty, a full bin is brought from the store room and an order is
placed for another bin.
Another model is designed to help evaluate those situations where the
inventory is perishable, such as with vegetables and newspapers.
Simulation models, as discussed in Production and Operations
Management courses, work well for these situations so we will not go
into them here.
We close this Module by examining the "MakeorBuy Analysis" Case on
pages 574575 in the text.
5.4: "A MakeorBuy
Analysis"
This case requires that we compare
making an inventoried item versus buying it from an outside supplier.
We will follow the numbered questions on page 575 in the text.
1. The holding cost is made of of four items, as shown below. Note
that I converted some of the items into percents for a common unit of
measurement. The holding cost applies to both making and buying
options,
Cost of Capital = 14%
Taxes/Insurance = $24,000/$600,000 = 4%
Shrinkage = $9,000/$600,000 = 1.5%
Warehouse Overhead = ($15,000/$600,000) = 2.5%
Total = 22%
2. The ordering cost applies to the "buy" option.
2 Hours at $28.00 = $56.00
Other Expenses ($2,375/125) = $19.00
Total = $75.00
3. The setup cost applies to the "make" option.
8 Hours at $50.00 = $400 per setup
4. and 5.
First, examine the "buy" option: order and buy from the supplier 
the EOQ Model. Assume in the case that one week = 5 days. The
output includes the pertinent inventory policy.
INVENTORY MODEL
***************
ORDER QUANTITYREORDER POINT WITH PROBABILISTIC DEMAND
******************************************************
YOU HAVE INPUT THE FOLLOWING DATA:
**********************************
ANNUAL DEMAND = 3200 UNITS PER YEAR
ORDERING COST = $75 PER ORDER
INVENTORY HOLDING COST:
A. ANNUAL INVENTORY CARRYING CHARGE = 22.0%
B. COST PER UNIT = $ 18 PER UNIT
WORKING DAYS PER YEAR = 250 DAYS
LEAD TIME DEMAND DISTRIBUTION IS NORMAL WITH
A. MEAN = 64 UNITS
B. STANDARD DEVIATION = 10 UNITS
PROBABILITY OF A STOCKOUT = 0.11
INVENTORY POLICY
****************
OPTIMAL ORDER QUANTITY 348.16
ANNUAL INVENTORY HOLDING COST $738.45
ANNUAL ORDERING COST $689.35
TOTAL ANNUAL COST $1,427.80
MAXIMUM INVENTORY LEVEL 360.56
AVERAGE INVENTORY LEVEL 186.48
REORDER POINT 76.40
NUMBER OF ORDERS PER YEAR 9.19
CYCLE TIME (DAYS) 27.20
SAFETY STOCK 12.40
ANNUAL SAFETY STOCK COST $49.10
EXPECTED STOCKOUTS PER YEAR 1.01
PROBABILITY OF A STOCKOUT PER CYCLE .1100
Note that I had to run the output twice to
get the probability of a stock out per order cycle. The first run
gave me the number of order cycles (number of orders) as 9.2. Since
the case information said that one stock out per year (one order
cycle will be out of stock) is acceptable, I then computed the
probability of a stock out as 1 / 9.2 = 0.11. This was input into
The Management Scientist to get the safety stock and safety
stock cost as shown above.
Note also that to get the Total Cost, we add the Total Inventory Cost
to the unit cost:
TC = TIC + D * C = $1427.80 + (3,200 * 18) = $59,029
Next, we examine the "make" alternative. I assumed an annual
production rate of 12,000, even though the case said that five months
of production capacity was available, I needed to annualize the 1,000
units of capacity a month. Since the demand is less than the
capacity, this assumption will work. The output includes the
inventory policy.
INVENTORY MODEL
***************
ECONOMIC PRODUCTION LOT SIZE
****************************
YOU HAVE INPUT THE FOLLOWING DATA:
**********************************
ANNUAL DEMAND = 3200 UNITS PER YEAR
ANNUAL PRODUCTION RATE = 12000 UNITS PER YEAR
SETUP COST = $400 PER SETUP
INVENTORY HOLDING COST:
A. ANNUAL INVENTORY CARRYING CHARGE = 22.0%
B. COST PER UNIT = $ 17 PER UNIT
WORKING DAYS PER YEAR = 250 DAYS
LEAD TIME FOR A NEW ORDER = 10 DAYS
INVENTORY POLICY
****************
PRODUCTION LOT SIZE 966.13
ANNUAL INVENTORY HOLDING COST $1,324.88
ANNUAL SETUP COST $1,324.88
TOTAL ANNUAL COST $2,649.76
MAXIMUM INVENTORY LEVEL 708.49
AVERAGE INVENTORY LEVEL 354.25
REORDER POINT 128.00
NUMBER OF SETUPS PER YEAR 3.31
CYCLE TIME (DAYS) 75.48
This module does not compute the cost of
safety stock. Since there are 3.3 production runs or cycles a year,
the probability of a stock out (one cycle out of stock) is 1 / 3. 3
or 30%. The Z score for 30% is 0.52. The reorder point is
then:
Reorder Point = 128 + Safety Stock
Reorder Point = 128 + Z * Std Dev_{Demand During Lead Time }Reorder Point = 128 + (.52 * 20) = 138
The annual inventory holding cost must be increased to recognize the safety stock. The maximum and average inventory levels also recognize the addition of 10 units of safety stock. The new holding cost is $1325 + (0.22 * 17 * 10 units safety stock) = $1,362. The revised total inventory cost for the "make" option is $1,362 + $1,325 = $2,687. This gives a new Total Cost of:
TC = TIC + (D * C) = $2,687 + (3,200 * $17) = $57,087.
From an economic point of view, it appears that Wagner Fabricating Company should make the item and save about 3%.


