Chapter 5 Inventory Management
Review Questions
1. It has been stated that, as a rule of thumb, ‘‘the best inventory is no inventory.’’ Discuss
this heuristic.
Answer: Inventory is an idle asset. Sitting around it produces nothing, and it costs to keep it.
We say it is non-value-adding. Thus, with less inventory kept in storage, there is less waste
and lower cost. This may be relatively valid in manufacturing, but it is unlikely to be true in
retailing. In retailing, inventory serves additional purposes, such as attracting customers to
displays. The heuristic contradicts an old marketing adage, “You can’t sell from an empty
wagon.” This heuristic must be applied to the right situation to be valid. Just in case we have
some readers who aren’t familiar with the meaning of heuristic, consider it to be a pragmatic
set of guidelines (for understanding a situation). Pragmatic is an important adjective because
it emphasizes rules based on experience rather than theory (i.e., a rule of thumb).
2. How does this rule of thumb apply to gasoline? Is gasoline an order point policy (OPP)
commodity?
Answer: While it does cost to keep gasoline in some form of storage, gasoline is not an
“idle asset.” A traditional automobile will not run without gasoline, and it is not possible to
know the exact amount of gasoline that will be needed to operate the automobile for the next
day or two. If an extra supply of gasoline is maintained in the garage, then it is possible that
this inventory is superfluous and even dangerous. Of course, this is not the case if hurricanes
occur regularly. Gasoline is an OPP commodity with characteristics of a perpetual inventory
system. Drivers are able to monitor their inventory of gasoline with fuel gauges. They will
purchase additional fuel whenever the inventory falls to the “reorder point” (perhaps when a
warning light comes on). However, there is no set economic order quantity; drivers either
purchase what they can afford at that time or fill up their tanks. This discussion becomes
even more involved when talking about hybrid engines which require some gasoline to
charge the batteries. Then, depending upon how that car is used (highway or local) refueling
can be very infrequent. Residue buildup can occur when fuel tanks are left barely filled for
long periods of time.
3. For a toothpaste manufacturer, how is the decision made concerning how many caps
should be ordered? Could it be a different number than the number of tubes that are
ordered at one time?
Answer: Over time, the number of caps should be about the same as the number of tubes.
But caps and tubes may come from different suppliers, or they may be made by processes
having significantly different economic run sizes. There is no reason for the order quantities
to be related. Here is an example: caps are ordered monthly, 52,000 at a time; tubes are
1
ordered weekly, 12,000 at a time. Over the year, the same quantity (624,000) of each (caps
and tubes) is used.
4. How often should an order size be updated?
Answer: Order sizes should be updated on a regular basis, such as in an annual review, even if there
is no obvious problem. This prevents policies from becoming outdated. Order sizes should also be
reviewed on an event basis, such as a change in supplier, and on an exception basis, such as when
interest rates, prices, or demand levels are no longer in the same range for which the order size
calculations were valid. Technological changes in manufacture and alterations of delivery conditions
may also affect order size. Since many of the above changes are occurring with greater frequency, an
annual review may be too long a period to wait.
5. What method should be used to determine the order quantity for a raw material that is used
continuously within a flow shop?
Answer: Flow shops use material on a regular and predictable basis; they tend to use the same
material for months or years. EOQ may be an appropriate model for flow shops if sometimes they
work overtime or consume variable amounts of the materials in line with other parameters.
6. What method should be used to determine the order quantity for a raw material that is used
continuously within an intermittent flow shop?
Answer: Intermittent flow shops fall between job shops and flow shops in their volumes of
output and in their continuity. Job shop outputs are not as uniform as flow shop outputs. It
would be foolish to use EOQ to schedule the arrival of raw materials for which there is no
job to use them. Job shops are more likely to order materials and parts on a per-job basis, so
that when a job is done, there are no leftovers. On the other hand, if the intermittent flow
shop has a high degree of regularity, running (say) 3 days a week, then it is possible to use
EOQ order policies. Finally, the intermittent flow shop (as epitomized by the EPQ model)
should forecast needs when the periods between runs are irregular.
7. Who are the people that are responsible for placing orders?
Answer: Purchasing agents and buyers place orders with external organizations. Production
schedulers requisition raw materials and parts in support of planned output. These may
include internal requests from other production departments as well as external request that
are funneled through purchasing agents. There are also orders to ship goods. Such orders
might come from warehouse managers, production schedulers, and marketing managers.
2
8. Benetton is a well-known manufacturer and retailer of clothing all over the world. The
Benetton factories are tied in with retailers so that demand information is relatively
immediate and complete concerning what is selling and what is not. How does this
information affect lot-size planning?
Answer: Lot-size planning is based in part on estimates of demand. As demand increases,
so does the lot size. Benetton’s wide variety of products includes many which may be
seasonal or faddish; they may be more concerned with timeliness in the production of
demanded items than with production quantities. The more timely data, coupled with a
faddish industry, suggest that manufacturing schedulers use a relatively short order horizon.
It should be adaptable to changes on short notice. Wherever 24/7 electronically generated
order data is used, ordering costs are reduced; smaller batch sizes are required; money is
saved and better timeliness is achieved.
9. Salespeople use handheld telecommunications devices to communicate inventory status to
the warehouse in a major toy company. Why is this system needed and what does it
affect?
Answer: The immediacy of the data reduces lead times, and therefore decreases variability
that must be dealt with. The electronic system speeds replacement (fill-in) sales, which may
prevent lost sales from stock-outs at the retail level. In the systems view, the organization
suffers if any part of this supply chain cannot execute the plan. Quality, demanded goods
cannot be sold if they are in the warehouse and meanwhile the retailer is out of stock. Unless
the toy is a classic with enduring demand, lost sales at retail translate into inventory that will
never leave the warehouse.
10. What is carrying cost composed of and what is the range of values that will be found for
this cost under varying conditions?
Answer: The major elements of carrying cost are opportunity costs for alternative uses of
funds (9 to 15 percent), theft (up to 6 percent), obsolescence (0 to 5 percent—but increasing
every year), deterioration (1 to 3 percent), and handling (1 to 2 percent) Minor elements in
carrying cost are taxes, storage, insurance, and miscellaneous (up to 0.25 percent each).
Interest rates have been almost negligible in the past few years, and company profits have
been volatile, so there might be reason to decrease the value assumed for opportunity costs.
11. What is the logic for a buyer to accept a discount offer?
Answer: Discounts save money but increase the number of units acquired. The reduction of
annual total purchase costs is represented by (c - c )*D. Acquisition cost savings are shown
3
by the equation for a variety of price breaks. The purchase price reduction frequently more
than offsets the increase in the carrying costs of greater stock levels. Usually, ordering costs
also decrease since more is being bought at one time because of the discount price. People
are accused of buying what they don’t need because of discounts. That should not occur for
business people who have excellent models for deciding when to accept a discount.
12. What is the logic for a seller to offer a discount?
Answer: The offered discount encourages fewer but larger orders, which is cost-efficient
from the supplier’s point of view. The supplier has its own set-up and carrying-cost factors,
which undoubtedly favor large orders. The discount offered to a customer is a way of
moving the customer’s purchase quantity closer to the optimum for the supplier’s production
and supply chain system.
13. Why is there an ordering cost and of what is it composed?
Answer: There is an ordering cost because the making and placing of an order uses scarce
resources. In the inventory policy model this cost is a fixed amount per order (regardless of
the size of the order). Ordering cost is the cost of labor (clerical and managerial, including
approvals), space, equipment, and materials used in making and placing an order. Costs that
are appropriate for inclusion in the ordering cost are those costs that are variable—with the
frequency of ordering (i.e., those that are not fixed costs of the purchasing department).
14. What is the difference between an ordering cost and a setup cost?
Answer: An ordering cost is the cost of making and placing an order with an external
supplier; a set-up cost is the cost of making and placing an order for self-supply (i.e.,
producing the item internally).
15. When is it likely that an ordering cost will be larger than a setup cost?
Answer: Ordering costs are likely to exceed set-up costs when set-up costs have been
dramatically reduced by use of good methods engineering techniques. There are also
reductions of setup costs due to group technology and flexible manufacturing processes.
Both can show numerous examples of reduced setup costs. Ordering costs, on the other
hand, are seldom scrutinized using efficiency improvement techniques. That presents an
obvious opportunity to find savings in operations that are taking place consistently and
everywhere.
4
16. When is it likely that a setup cost will be larger than an ordering cost?
Answer: Setup costs are often process changeover costs which include lost output and
product wasted in set-up and testing. These costs can be considerable. Ordering costs are
likely to be less than set-up costs if the vendor-customer link uses EDI (electronic data
interchange), or the Internet. Trust arrangements between certified vendors and customers
can result in pre-agreed upon delivery schedules (e.g., vendor releasing).
17. Relate the number of orders placed and the order quantity.
Answer: The number of orders placed n (per time period) times the order quantity Q equals the
demand (per time period). That is: Q  n = D.
18. Why are order point policies (OPP) called by that name?
Answer: Order Point Policies (OPP) are so named because the event that triggers a
replenishment order is when the inventory level reaches the “reorder point.” The name is
assigned because the reorder point is the constant element—the timing between orders is
variable.
19. Why are total variable cost equations written for OPP models instead of total cost
equations?
Answer: Total cost equations include carrying cost, ordering or set-up cost, and acquisition
cost. In OPP models there is no quantity discount to consider, so that c  D is a constant.
Any cost that is a constant regardless of the decision reached is not relevant to the decision.
Therefore, OPP models can safely remove acquisition cost from the data used for decisions.
This leaves total variable cost (TVC) equations, which are the sum of carrying costs and
ordering or set-up costs. Both of the latter costs vary with order quantity.
20. When discounts are being considered, total cost equations must be used instead of total
variable cost equations. Why is this so?
Answer: When discounts are being considered, the acquisition cost c  D is no longer
constant. Different order quantity decisions will lead to different values for c  D. The
discount prices are denoted as c’< c, c”< c’, etc. Since the acquisition cost is not constant,
it must be included in the discount decision. In fact, if acquisition cost were not included in
the order quantity determination, the discount would never be taken, since any deviation
from EOQ raises the sum of carrying costs and ordering costs.
5
21. Differentiate between EOQ and EPQ models.
Answer: EOQ assumes instantaneous delivery of all of the units ordered for the product. Whereas,
EPQ assumes that the ordered product arrives in a smooth manner over a period of time (for
example, so many units arrive per hour or per day). This smooth arrival rate, coupled with some
usage rate, means that some items are used directly and never enter inventory. This reduces the
average inventory on which carrying costs are calculated. This lower carrying cost leads to a larger
lot size for ELS than for a comparable EOQ model. These two patterns of arrivals for stock are
usually associated with vendors for EOQ and self-supply for EPQ.
22. When is lead-time variability a problem and what can be done about it?
Answer: Variable lead times (LT) compound the probabilistic nature of variable demand.
They add an extra source of uncertainty. Variable LT complicate the process of controlling
stock-outs and calculating buffer stocks. Variable lead time can be dealt with by better
partnership relations with suppliers, use of more reliable shipping firms, and expediting. If
variability cannot be reduced, then extra safety stock needs to be carried. Variability in any
form adds extra costs to the system’s management.
23. How can lead-time variability be modeled?
Answer: Variable lead time and variable demand taken together constitute a joint distribution of two
variables. This can be modeled with equations but the formulations are difficult. It is possible to sum
the variances of the two distributions in an effort to estimate the amount of variability that these two
sources of variation contribute. A simulation can be created with the joint distributions—and that
approach holds a lot of promise in understanding and coping with the larger degree of variability. It
is often useful to employ simplifying assumptions such as a “worst case” lead time. Then there are
those who use the ostrich management technique (ignore LT distributions).
24. What is a two-bin system? When is it applicable?
Answer: The “two-bin” system is a simple version of OPP modeling. Two “bins” are used
to hold an item of merchandise. Bin-One is filled to the reorder point. Bin-Two has the rest
of the inventory. Orders are filled from Bin-Two first. When it is emptied, a replenishment
order is generated. The reorder point has been reached. This system is easy to understand
and to implement; it generates little paperwork. It works well on small, low-value items such
as nuts and bolts, and liquids.
25. Describe a perpetual inventory system.
6
Answer: A perpetual inventory system continuously records withdrawals. Often, perpetual
inventory is associated with order point policies, where an order for a fixed amount is placed
whenever inventory falls to the reorder point. Continuous monitoring and awareness of the
level of inventory is essential for the concept of reorder points to be effective.
26. Under what circumstances is a perpetual inventory system preferred?
Answer: Perpetual inventory is preferred when inventory position is important enough to be
known at all times. This is when the value of knowing inventory position outweighs the
added cost of monitoring and checking. When the costs of operating the system are very
high, we may not select the perpetual inventory model (see Review Question 27). Perpetual
models are used for high-value items, items critical to operations, or items that are timesensitive (short shelf lives, for example). Also, as computer systems have become less
expensive and more powerful, the cost of using the perpetual system has decreased to the
point where it is preferred overall except for C-type items.
27. Describe a periodic inventory system.
Answer: In a periodic inventory system, an item’s inventory position is checked at fixed
time intervals—the “period” of the policy’s name. The time interval between orders is fixed,
but the amount of the order is variable. This happens because the amount in stock will differ
at each periodic inspection—the result of variation in demand. The size of the order is the
difference between stock on hand and the calculated target inventory. This system requires
keeping track of stock withdrawals but only needing to subtract withdrawals on the day as
specified by the period (e.g., every other Monday, check stock on paper towels for the spa).
28. When is a periodic inventory system preferred?
Answer: A periodic inventory system is preferred for low-value items (C-type) or items not
critical to operations. Periodic is preferred for manual inventory systems because it requires
less record-keeping and paperwork. Small business owners may not have the time or money
to use perpetual systems. Periodic inventory may be preferred when multiple products need
to be ordered together—the fixed time interval coordinates that activity. Periodic inventory
may be required to coordinate production changeovers, or may be necessitated by a supplier
who accepts orders only on certain days or where deliveries are scheduled only on specific
days. Periodic inventory systems have become less popular as point-of-sale systems and
7
other computerized transaction systems have made perpetual inventory much less expensive
to use.
29. How can a quantity discount model be used by a buyer and supplier to negotiate a price
break point schedule that benefits both of them?
Answer: Quantity discounts may benefit both parties. Customers save more on acquisition
costs than they pay extra for inventory carrying and ordering costs. Suppliers save more on
fewer set-ups than they give up on sales revenues (which are the result of the customer’s
acquisitions). The quantity discount model is based on an analysis of total costs. Discounts
are offered voluntarily by suppliers, and taken (or rejected) voluntarily by customers. Both
parties can calculate the impact of any proposed discount schedule on their total costs, and
will know how advantageous (or not!) that discount might be. This is the basis for
negotiating a mutually advantageous discount.
30. Why is it that multiple price break points for many discount levels can be examined in the
same fashion as one price break point for a single discount?
Answer: There is no logical distinction between two price levels and more than two. Each
price level generates an inventory cost curve. But for any specified quantity, only one price
level is feasible, as determined by the schedule that relates discounts to quantities purchased.
The optimal decision is reached by finding the point of lowest total cost on these feasible
sections of the cost curves. It does not matter (except for more complex arithmetic) whether
there are two (or more) feasible segments. It is essential, however, to know that the lowest
possible total cost has been selected by the acceptance of any discount offered. If the amount
ordered is very large, it behooves the buyer to examine the costs of obsolescence, spoilage,
etc.
31. Is the two-bin inventory system perpetual or periodic?
Answer: The two-bin model is not periodic because there is no inspection at any fixed time
interval. It is an order point model in that the emptying of one bin (a quantity event, not a
time event) triggers replenishment. Perpetual inventory is a type of order point model. The
two-bin model is labeled a perpetual model, but there is no need for continuous recording of
inventory position; in fact, it is a near-paperless inventory model.
32. Distinguish between inventory problems under certainty, risk, and uncertainty.
Answer: Certainty problems have known demands and lead times. Supplier contracts are an
example. There is no need to employ probability distributions. Risk problems have variable
8
demands and/or lead times—for which probability distributions must be established.
Uncertainty problems have no probability distributions available. One must assume the
nature of the uncertainty. For example, a uniform distribution states that all possibilities are
equally likely. Risk and uncertainty add sources of variation to a situation and therefore add
to cost. The basic EOQ and EPQ models are models of certainty—there are no probabilistic
factors to consider. Order point models and periodic review models are models of risk—the
normal distribution is used in the calculation of buffer stock or target inventory. An example
of uncertainty would be the Christmas tree problem with no available probabilities for
demand. Such a problem would be solved by decision table tools (without calculating
expected values). For example, it is possible to minimize regret.
9
Problems
Problem 1
1. Water testing at the Central Park reservoir requires a chemical reagent that costs
$500 per gallon. Use is constant at 1/3 gallon per week. Carrying cost rate is
considered to be 12 percent per year, and the cost of an order is $125. What is the
optimal order quantity for the reagent?
Answer
8.50
EOQ Model
Yearly Demand (D)
Order Cost/Order (S)
Inv Cost/unit/year (i)
Item Cost (C)
17.33
$125.00
12.00%
$500.00
=52*(1/3)
Inv Cost/unit/year (H)
$60.00
=0.12*500
EOQ
8.50
EOQ =
Total Cost (TC)
TC = (D/Q)*S + (Q/2)*H +
D*C
$9,176.57
10
Problem 2
2. Continuing with the information about the Central Park reservoir given in
Problem 1, the city could make this reagent at the rate of 1/8 gallon per day, at a
cost of $300 per gallon. The setup cost is $150. Use a 7-day week. Compare using
the EOQ and the EPQ systems. What course of action do you recommend?
Answer
15.28
EPQ Model
Yearly Demand (D)
Setup Cost/Order (S)
Inv Cost/unit/year (i)
Item Cost (C)
Inv Cost/unit/year (H)
17.33
$150.00
12.00%
$300.00
$36.00
Production Rate per day (p)
0.125
Demand Rate per Day (d)
EPQ
0.048
15.28
=52*(1/3)
=0.12*300
EPQ =
Total Cost (TC)
$5,540.42
TC = (D/Q)*S + (Q/2)*H*(1-d/p) +
D*C
Answer: Comparing with total cost in problem 1, there is a saving of $3,636.15
($9,176.57 - $5,540.42).
11
Problem 3
3. Water testing at the Delaware reservoir requires a chemical reagent that costs
$400 per gallon. Use is constant at 1/3 gallon per week. Carrying cost rate is
considered to be 10 percent per year, and the cost of an order is $100. What is the
optimal order quantity for the reagent?
Answer
9.31
EOQ Model
Yearly Demand (D)
Order Cost/Order (S)
Inv Cost/unit/year (i)
Item Cost (C)
Inv Cost/unit/year (H)
EOQ
17.33
$100.00
10.00%
$400.00
$40.00
9.31
Total Cost (TC)
TC = (D/Q)*S + (Q/2)*H + D*C
$7,305.71
12
=52*(1/3)
=0.12*500
EOQ =
Problem 4
4. Continuing with the information about the Delaware reservoir given in Problem
3, the town could make this reagent at the rate of 1/4 gallon per day, at a cost of
$500 per gallon. The setup cost is $125. Use a 7-day week. Compare using the
EOQ and the EPQ systems. What course of action do you recommend?
Answer
10.35
EPQ Model
Yearly Demand (D)
Setup Cost/Order (S)
Inv Cost/unit/year (i)
Item Cost (C)
Inv Cost/unit/year (H)
17.33
$125.00
10.00%
$500.00
$50.00
Production Rate per day (p)
0.25
Demand Rate per Day (d)
EPQ
0.048
10.35
Total Cost (TC)
TC = (D/Q)*S + (Q/2)*H*(1-d/p) + D*C
=52*(1/3)
=0.10*500
EPQ =
$9,085.47
Answer: Comparing with total cost in problem 3, there is a saving of $1,779.76
($9,085.47 - $7,305.71) if purchased from outside.
13
Note: See Excel file SMCh05 for calculations of problems 5, 6 and 7.
Problem 5
5. The Drug Store carries Deodorant R, which has an expected demand of 15,000 jars per year
(or 60 jars per day with 250 days per year). Lead time from the distributor is three days. It has
been determined that demand in any 3-day period exceeds 200 jars only once out of every 100 3day periods. This outage level (of 1 in 100 LT periods) is considered acceptable by P/OM and
their marketing colleagues. The economic order quantity (EOQ) has been derived as 2,820 jars.
Set up the perpetual inventory system (EOQ with stock-outs ).
Solution:
Order 2,820 jars when the inventory level hits 200 jars which is the chosen reorder point (ROP).
This ROP provides the acceptable outage level of one occurrence in a hundred, i.e., 0.01. Next,
we note that the 200 units of the designated ROP are composed of: first, the sum of the expected
demand in any 3-day lead time period which is 180 units, and second, 20 units of Buffer Stock
(BS). In summary, we have estimated BS = 20 units with the chosen ROP = 200 units and this
yields the acceptable outage rate of 1/100 LT periods. We have determined BS = zσ = 20. Refer
to the z-table where the lookup value for the tail of 0.01 provides the z-value of 2.325. From this
we find that the implied value of σ is 20/2.325 =8.602. The variance σ2 = 74 and sigma for a day
is equal to the square root of 74 which equals 4.97. This gives all necessary information to set up
the perpetual inventory system (see sections 5.10.1 and 5.10.2).
Problem 6
6. Use the information given in Problem 5, plus the fact that it has been determined that demand
in any 50-day period exceeds 4,000 jars only once out of every 100 50-day periods. This
outage level (of 1 in 100 LT periods) is considered acceptable by P/OM and their marketing
colleagues. Set up the periodic inventory system. (EOQ with stock-outs.)
Solution:
The optimal review interval is that period of time required to use up the EOQ stock level. Since
the demand of 15,000 units/year is 15,000/250 = 60 units/day, there are 2,820/60 = 47 days
between reviews. The lead time for replenishment is 3 days. With the periodic model, the time
from placing an order to its arrival is 47 + 3 = 50 days. This includes the periodic time interval of
47 days between reorders.
The chance of demand being greater than 4,000 over this 50-day period is (a satisfactory) one
percent. The M-level for the periodic model is set at 4000. See Figure 5.12 where the M-level is
shown as the horizontal line on the chart. Demand in the reorder period is 2820 units and demand
in the lead time period is 180 units. Then, we calculate 4000 – 2820 – 180 = 1000 units which is
a reasonable estimate for buffer stock (BS). Note: M is set by adding EOQ + D(LT) + BS. Thus,
14
when the reorder quantity at the periodic review time is the same as the EOQ, it is because all the
units of inventory except BS have been depleted. Also, we have determined BS = zσ = 1000.
With z = 2.325 as in Problem 5, σ = 1000/2.325 = 430.11.
See the spreadsheet SMCh05. Problems 6 estimates the standard deviation for M = 4000 units at
430 units. This does not seem out of line. Also quite reasonable for the periodic model is the high
daily sigma of demand. It is 60.83. The periodic system requires large buffer stocks to protect
against out-of-stocks caused by demand variability over the long period of 50 days between
replenishments.
Problem 7
7 Compare the results derived in Problems 5 and 6. What do you recommend doing? Explain
how you have taken into account all of the important differentiating characteristics of
perpetual and periodic inventory systems. (EOQ with stock-outs.)
Solution:
The perpetual system must protect against outages for the short period of time that is the lead
time—not the period between reviews. Since the perpetual is less wasteful than the periodic
system, why ever use the periodic. First, when the delivery system is geared to periodicity, (i.e.,
the ship leaves on the first of each month) the perpetual cannot be used. Second, before
computers, computations were costly. Many people would have been involved in maintaining an
accurate perpetual total. Therefore, for C-type and even B-type items, the periodic system was a
less expensive way to go. Comparing Problems 5 and 6, note how much smaller the perpetual
buffer stock size is than the periodic buffer size (20 to 1,000). The comparison is 50:1. Perpetual
needs smaller amounts of storage space, there is less damage and pilferage, there are lower
amounts of cash tied up in holding costs. In general, opportunity cost savings of the perpetual are
evident, and the system’s managers enjoy much smaller inventory fluctuations. The spreadsheet
shows that sigma for the periodic system is more than twelve times greater than for the perpetual.
Problem 8
8. Consider the recommendation made in Problem 5, taking into account the fact that The Drug
Store must combine orders for Deodorant R with other items in order to have sufficient volume
to qualify for the distributor’s shipping without charge. With this constraint, what are your
recommendations?
Solution:
If orders must be combined to save on shipping charges, then a periodic inventory system is
indicated. The shipping costs dominate the decision when they are larger than the savings
obtained by using the optimal order quantity of the perpetual inventory model instead of the
15
optimal order interval of the periodic inventory model. Thus, various items, including Deodorant
R, will be ordered every (say) 30 or 40 days. The optimal period of 47 days for Deodorant R has
to be altered to fit other items as well. Using judicious consideration, a compromise period must
be determined. On the other hand, using a systems approach, it is necessary to investigate other
methods of shipping including UPS and FedEx (with special arrangements) and the USPS which
has been making all kinds of special deals with Amazon, Wal-Mart and other large retailers.
Problem 9
9. The information required to solve an EOQ inventory problem is as follows:
demand per year (D) = 5,000, ordering cost per order (S) = $ 10.00, Cost of the
item (C) = $10 per unit, and inventory carrying cost per unit per year (H) = 16% of
the cost of the item (C). What is the optimal order quantity? (EOQ model)
Answer
250.00
EOQ Model
Yearly Demand (D)
Order Cost/Order (S)
Inv Cost/unit/year (i)
Item Cost (C)
Inv Cost/unit/year (H)
5000.00
$10.00
16.00%
$10.00
$1.60
EOQ
250.00
=0.16*10
EOQ =
Total Cost (TC)
TC = (D/Q)*S + (Q/2)*H +
D*C
$50,400.00
16
Problem 10
10. Using the information in Problem 9, instead of buying from a supplier the
decision is to make the item in the company’s factory. The new equipment is able
to produce 30 units per day. Cost per item is now $6.00. The set up cost is $ 150.00
per set up. What is the optimal run size (EPQ)?
Working Days per Years
Answer
EPQ Model
Yearly Demand (D)
Setup Cost/Order (S)
Inv Cost/unit/year (i)
Item Cost (C)
Inv Cost/unit/year (H)
Production Rate per day (p)
Demand Rate per Day (d)
EPQ
Total Cost (TC)
TC = (D/Q)*S + (Q/2)*H*(1-d/p) +
D*C
250
2165.06
5000.00
$150.00
16.00%
$6.00
$0.96
Assumed
=0.16*6
30
20.000
2165.06
EPQ =
$30,692.82
Run Time (Cycle Time)
Maximum Inventory Level
72.17
721.69
=EPQ/d
=2165/30
Imax = Q*(1- d/p)
Average Inventory
360.84
(Imax)/2
The optimal run size with self-supply is more than 8 times the optimal order
quantity (EOQ = 250).
17
Problem 11
11. Using the information in Problems 9 and 10, which is better: make or buy?
Answer
Make
Comparing with total cost in problems 9 and 10, there is a saving of $19,707.18
($50,400 - $30,692.82) if the item is made in house.
Problem 12
12. Using the information in Problems 9—11, what factors that are not part of the mathematical
model might shift the decision?
Solution:
Additional factors not in these equations that might shift the decision from make to buy include:
unexpectedly large training costs to develop the necessary skills, low capacity problems that
make it impossible to match supply and demand, finding new suppliers that can produce items at
lower costs than previously thought possible, and production difficulties encountered that were
not anticipated.
18
Problem 13
13. The following quantity discount schedule has been offered for the situation described in
Problem 9. Should either of these discounts be accepted?
$ 10.00 for Q ≤ 299
$ 9.00 for 300 ≤ Q ≤ 499
$ 8.00 for Q ≥ 500
Data from problem 9.
The information required to solve an EOQ inventory problem is as follows: demand per year
(D) = 5,000, ordering cost per order (S) = $ 10.00, Cost of the item (C) = $10 per unit,
and inventory carrying cost per unit per year (H) = 16% of the cost of the item (C). What is
the optimal order quantity? (EOQ model)
EOQ with Price = $ 8.00
Yearly Demand (D)
5000.00
Order Cost/Order (S)
$10.00
Inv Cost/unit/year (i)
16.00%
Item Cost (C)
$8.00
Inv Cost/unit/year (H)
$1.28
=0.16*8
EOQ
279.51
This EOQ (279.51)<500. Therefore, EOQ =
it is not feasible.
Best Quantity for price ($ 8.00) =
500
Total Cost at Q = 500
$40,420.00
TC = (D/Q)*S + (Q/2)*H + D*C
EOQ with Price = $ 9.00
Yearly Demand (D)
5000.00
Order Cost/Order (S)
$10.00
Inv Cost/unit/year (i)
16.00%
Item Cost (C)
$9.00
Inv Cost/unit/year (H)
$1.44
=0.16*9
EOQ
263.52
This EOQ (263.52)<300. Therefore, EOQ =
it is not feasible.
Best Quantity for price ($ 9.00) =
300
Total Cost at Q = 300
$45,382.67
TC = (D/Q)*S + (Q/2)*H + D*C
EOQ with Price = $ 10.00
Yearly Demand (D)
5000.00
Order Cost/Order (S)
$10.00
Inv Cost/unit/year (i)
16.00%
Item Cost (C)
$10.00
Inv Cost/unit/year (H)
$1.60
EOQ
250.00
=0.16*10
EOQ =
Total Cost (TC)
$50,400.00
TC = (D/Q)*S + (Q/2)*H + D*C
Answer: Q = 500 is the best quantity because it minimizes
the total cost.
19
Problem 14
14. The quantity discount schedule offered in Problem 13 prompted a competitor to offer the
following discount schedule. Should any of these discounts be accepted?
$ 10.00 for Q ≤ 250
$ 9.00 for 251 ≤ Q ≤ 599
$ 7.00 for Q ≥ 600
Data from problem 9.
Demand per year (D) = 5,000, ordering cost per order (S) = $ 10.00, Cost of the item (C) =
$10 per unit, and inventory carrying cost per unit per year (H) = 16% of the cost of the item
(C). What is the optimal order quantity? (EOQ model)
EOQ with Price = $ 7.00
Yearly Demand (D)
5000.00
Order Cost/Order (S)
$10.00
Inv. Cost/unit/year (i)
16.00%
Item Cost (C)
$7.00
Inv. Cost/unit/year (H)
$1.12
=0.16*7
EOQ
298.81
This EOQ (298.81) < 600. Therefore, it EOQ =
is not feasible.
Best Quantity for price ($ 7.00) =
600
Total Cost at Q = 600
$35,419.33
TC = (D/Q)*S + (Q/2)*H + D*C
EOQ with Price = $ 9.00
Yearly Demand (D)
5000.00
Order Cost/Order (S)
$10.00
Inv. Cost/unit/year (i)
16.00%
Item Cost (C)
$9.00
Inv. Cost/unit/year (H)
$1.44
=0.16*9
EOQ
263.52
This EOQ is feasible.
EOQ =
Total Cost (TC)
$45,379.47
TC = (D/Q)*S + (Q/2)*H + D*C
Answer: Q = 600 is the best quantity because it minimizes the total cost.
Note: In this problem there is no need to calculate EOQ at $ 10.00 since a
feasible quantity has been found at a price of $ 9.00.
20
Problem 15
15. Compare the answers to Problems 13 and 14 and discuss these results, making appropriate
recommendations.
Solution:
In the previous two problems, unit costs dictate the decision. In Problem 14, buying in lots of
600 from the competitor for C = $7 has a total unit cost of $35,000. That is considerably less
than the $50,000 at C = $10 or even $45,000 at C = $9. The competitor in Problem 14 beats the
competitor in Problem 13 by $5,000 ($40,420.00 - 35,419.33). See the total cost calculations in
Problems 13 and 14.
21
Problem 16
16. Murphy’s is famous for their coffee blend. The company buys and roasts the
beans and then packs the coffee in foil bags. It buys the beans periodically in
quantities of 120,000 pounds and assumes this to be the optimal order quantity.
This year it has been paying $2.40 per pound on a fairly constant basis. The
company ships 1,200,000 1-pound bags of its blended coffee per year to its
distributors. This is equivalent to shipping 24,000 1-pound foil bags in each of 50
weeks of the year. This can be considered to be constant and continuous demand
over time. What carrying cost in percent per year is implied (or imputed) by this
policy if an order costs $100 on the average? Discuss the results
In this problem, find the value of inventory carrying coast per unit per year (H),
given D, S and EOQ.
The value of H will be = 2DS(Q^2)
Q=
Yearly Demand (D)
Order Cost/Order (S)
Item Cost (C)
Order Quantity (Q)
1200000.00
$100.00
$2.40
120000.00
Inventory Carrying Cost per
Unit per year
$0.02
Inventory Carrying Cost as a
% of Item Cost
0.7%
= H/C
Answer: The implied carrying cost is 0.7% which is rather low. The model does not
consider fluctuations in coffee market prices, which may determine Murphy’s real
buying pattern during the year. If the price of coffee could be cut in half, the
carrying cost would double to 1.4 percent. Even so, the percentages are too low. It
is apparent that Murphy’s is not following an optimal ordering policy.
22
Problem 17
17. Using the information in Problem 16, suggest a better ordering policy.
To answer this question a more reasonable inventory carrying cost is to be assumed
and then the total cost will be compared for the current policy (order size = 120,000)
and economic order quantity.
Inv. Cost/unit/year (i)
Yearly Demand (D)
Order Cost/Order (S)
Inv. Cost/unit/year (i)
Item Cost (C)
Inv. Cost/unit/year (H)
12%
Calculations for the EOQ
1200000.00
$100.00
12.00%
$2.40
$0.29
EOQ
Assumed
=.12*2.4
28867.51
EOQ =
Total Cost (TC)
TC = (D/Q)*S + (Q/2)*H + D*C
$2,888,313.84
Calculations for the Current Policy
Yearly Demand (D)
1200000.00
Order Cost/Order (S)
$100.00
Inv Cost/unit/year (i)
12.00%
Item Cost (C)
$2.40
Inv Cost/unit/year (H)
$0.29
Order Size (Q)
0.288
120000.00
EOQ =
Total Cost (TC)
TC = (D/Q)*S + (Q/2)*H + D*C
$2,898,280.00
Answer: The current policy with a total cost of $ 2,898,280.00 is more expensive as
compared with EOQ policy with a total cost of $ 2,888,313.84.
Note: The two policies can be compared for various values of the inventory carrying
cost just by changing the assumed value of (i) in the above spreadsheet.
23
Problem 18
18. The manager of the greeting card production department has been buying two
rolls of acetate at a time. They cost $200 each. Card production requires 10 rolls
per year. Ordering cost is estimated to be $4 per order. What carrying cost rate is
imputed? Is it reasonable?
In this problem, find the value of inventory carrying coast per unit per year (H),
given D, S and EOQ (Q).
The value of H will be = 2DS(Q^2)
Q=
Yearly Demand (D)
Order Cost/Order (S)
Item Cost (C)
Order Quantity (Q)
10.00
$4.00
$200.00
2.00
Inventory Carrying Cost per
Unit per year
$20.00
Inventory Carrying Cost as a
% of Item Cost
10.0%
= H/C
Answer: The implied carrying cost is 10% which is reasonable.
24
Problem 19
19. Using the information in Problem 18, if the cost of rolls of acetate increases to
$250 each, what happens to the imputed carrying cost rate? Is this reasonable?
In this problem, find the value of inventory carrying coast per unit per year (H),
given D, S and EOQ (Q).
The value of H will be = 2DS(Q^2)
Q=
Yearly Demand (D)
10.00
Order Cost/Order (S)
$4.00
Item Cost (C)
$250.00
Order Quantity (Q)
2.00
Inventory Carrying Cost per
Unit per year
$20.00
Inventory Carrying Cost as a
% of Item Cost
8.0%
= H/C
Answer: A carrying rate of 8 percent is imputed, which is reasonable. If the real
economic conditions with respect to interest rates and the company’s ability to use
funds are known, then the imputations can be better corroborated.
Problem 20
20. In an inventory control system, the annual demand is 12,000 units; the
ordering cost is $30 per order and the inventory holding cost is $ 3.00 per unit
per year. The order quantity is 1000 units and the cost per unit of the item is
$150? What is the total cost (TC) per year? TC includes the inventory holding
cost, ordering cost and the cost of the item.
EOQ Model
Yearly Demand (D)
Order Cost/Order (S)
Item Cost (C)
Inv Cost/unit/year (H)
12000.00
$30.00
$150.00
$3.00
25
EOQ
Total Cost (TC)
TC = (D/Q)*S + (Q/2)*H + D*C
1000.00
$1,801,860.00
26
Problem 21
21. A company is planning for its financing needs. What is the total cost (TC)
per year given an annual demand of 12,000 units, setup cost of $32 per order, a
holding cost per unit per year of $4, an order quantity of 400 units, and a cost
per unit of inventory of $150? TC includes the yearly set up cost, inventory
holding cost and the item cost.
EOQ Model
Yearly Demand (D)
Order Cost/Order (S)
Item Cost (C)
Inv Cost/unit/year (H)
EOQ (Q)
Total Cost (TC)
TC = (D/Q)*S + (Q/2)*H + D*C
12000.00
$32.00
$150.00
$4.00
400.00
$1,801,760.00
27
Problem 22
22. If annual demand is 50,000 units, the ordering cost is $25 per order and the
holding cost is $5 per unit per year, what is the optimal order quantity?
Answer
707.11
EOQ Model
Yearly Demand (D)
Order Cost/Order (S)
50000.00
$25.00
Inv Cost/unit/year (H)
$5.00
EOQ
707.11
EOQ =
Problem 23
23. A company is using the Economic Order Quantity (EOQ) model to manage its inventories.
Suppose its annual demand doubles, while the ordering cost per order and inventory holding cost
per unit per year do not change. What will happen to the EOQ?
Answer:
EOQ =
If the annual demand doubles, then increase D to “2D” in the above equation.
New EOQ = 1.41 * Old EOQ.
The number 1.41 is square root of 2.
The students can test it by using numerical numbers.
28
Problem 24
24. Find the economic production quantity for the following problem. Also Identify the
storage capacity required.
Annual Demand = 50,000 units; Setup Cost = 25; Inventory Holding Cost = 5 per unit per
year.
Production rate = 500 units per day; number of working days = 250.
Working Days per Years
Answer
250
912.87
Storage Capacity
547.72
Assumed
EPQ
Max. Inv.
Level
EPQ Model
Yearly Demand (D)
Setup Cost/Order (S)
Inv Cost/unit/year (H)
50000.00
$25.00
$5.00
Production Rate per day (p)
500
Demand Rate per Day (d)
EPQ
Maximum Inventory Level
Note: Storage capacity is equal to the maximum inventory
level.
200
912.87
547.72
=EPQ*(1(d/p))
Maximum Inventory Level
29
EPQ =
Problem 25
25. Consider the following data answer the next four questions.
A plant operates 5 days a week, 52 weeks a year and can produce at 60
units per day.
o The setup cost for production run is $ 450.00.
o The cost of holding inventory is $ 5.00 per unit per year.
o The annual demand for this product is 5,000 units.
a. What is the Economic Production Quantity (EPQ)? 1150.88
Working Days per Years
EPQ Model
Yearly Demand (D)
Setup Cost/Order (S)
Inv Cost/unit/year (H)
Production Rate per day (p)
Demand Rate per Day (d)
EPQ
260
5 days/52 Weeks
5000.00
$450.00
$5.00
60
19.231
1150.88
EPQ =
b. What is the total annual cost of inventory and set up if the batch size (Q) is
800 units?
Batch Size (Q)
800
Maximum Inventory Level
543.59
Imax = Q*(1- d/p)
Average Inventory
Annual Inventory Cost
Number of Set-ups
Annual Set-up Cost
Total Annual Inventory and
Set-up Cost
271.79
(Imax )/2
$1,358.97
6.25
$2,812.50
H*(Imax )/2
D/Q
(D/Q)*S
$4,171.47
c. What is the number of set ups per year if the lot ize (Q) is 1500 units?
Lot (batch) Size
1500
Number of Set-ups
3.33
= 5000/1500
d. What is the maximum inventory level if the batch size (Q) is 600?
Batch Size
600
Maximum Inventory Level
407.69
Imax = Q*(1- d/p)
30
31