DECISION MODELING WITH
MICROSOFT EXCEL
Chapter 7
Nonlinear
Optimization
Part 3
Copyright 2001
Prentice Hall Publishers and
Ardith E. Baker
The EOQ Inventory Model
Inventories are defined as _________in storage,
waiting to be used. For example, inventories of
Raw materials
In-process materials
Finished goods
Cash
Individuals
Inventories are held for many reasons:
1. Inventories smooth out the ___________
between supply and demand.
2. The possibility of holding inventory often
contributes to lower _____________costs.
3. Inventories provide a way of ______labor.
4. Inventory is a way of providing quick
_________________at the time an item is
needed.
The costs associated with inventory activity are
_______costs, ordering costs, and _______costs.
STECO Example:
STECO stocks a short length optimal fiber
network cable (NC) used to connect Internet
routers to local area network equipment.
Consider the following costs:
Holding Costs:
3000 NCs x $8 per unit cost = $24,000
Note that this money is tied up in_________. By
holding inventory, STECO forgoes the opportunity
to make other investments (_______________).
Other components of holding costs include
_________, pilferage, insurance, warehousing,
and special handling requirements.
As inventory increases, holding cost__________.
Ordering Costs: Every time an order is placed
(independent of the ___________ordered), an
ordering cost is incurred. This cost is related to
the amount of ____________time required for
accounting, invoicing, order checking, etc. when
an order is placed.
Stockout Costs: When a company runs out of
________, a stockout occurs (i.e., orders arrive
after inventory has been depleted. There are two
ways to treat such orders:
1. ___________(save up the orders and fill
them later after the inventory comes in).
In addition to backlogging cost, stockout
cost includes the ________from late
delivery of stock.
2. No Backlogging
In this case, a _____________(the per unit
cost of unsatisfied demand) occurs.
Stockout cost includes the lost profit from
not making the__________.
Stockout cost can also include the cost of losing
the___________, loss of goodwill, and of
establishing a poor ________of service.
So, to avoid stockouts (and stockout costs), have
enough inventory_________. However, carrying
inventory implies a holding cost. This cost can be
reduced by _________more often. However, this
increases the ordering cost. It is important to
__________these three costs against each other.
For every type of ___________ordered, there are
two key questions that must be answered:
1. ________should an order be placed?
2. ________should be ordered?
Some considerations are:
The extent to which future _______is known.
The cost of ____________and management’s
policy (backlogging or not).
The inventory holding and ordering costs.
The possibility of long __________(the time
from when an order is placed to when it is
received).
The possibility of quantity___________.
STECO Wholesaling: The Current Policy
The monthly demand (orders received) is:
MONTH
DEMAND (UNITS)
January
February
March
April
May
June
July
August
September
October
November
December
Total Annual Demand:
Average Monthly Demand:
5,300
5,100
4,800
4,700
5,000
5,200
5,300
4,900
4,800
5,000
4,800
5,100
60,000
5,000
Over a period of several years, the demand has
remained at a steady rate of about 5000 NCs per
month.
STECO’s policy last year was to add 5000 NCs to
inventory each month. Since demand is expected
to be the same, this is also the current policy.
Based on this policy, assume
Shipments always arrive on the first day of
each month.
Demand is known and constant (at a rate of
5000 units per month).
No backlogging (i.e., no stockouts).
Plot of inventory on hand at any time:
5,000
Average
Inventory
2,500
0
1
2
3
4
5
6
Time
Given the previous_______________, the cost of
operating the system above depends only on how
much new stock is ordered and on the holding
and ___________costs.
Since demand is 5000 NCs and we order 5000
every month, the ___________inventory is
5000/2 or 2,500 NCs.
The effect of ordering 10,000 NCs every other
month is shown below.
10,000
Average
Inventory
5,000
0
1
2
3
4
5
6
Time
The demand remains __________at 5000 NCs
per month. Average inventory is doubled but the
annual number of orders is cut in________.
A policy of increasing the order quantity increases
the _________costs and decreases the annual
ordering cost.
Developing the EOQ Model:
The Economic Order Quantity (EOQ) model is a
simple model which attempts to ________the
cost of placing orders with the cost of holding
inventory.
The EOQ model assumes
No __________are allowed.
There is a constant rate of____________.
The __________costs are ordering and
holding costs.
The EOQ finds the _____________________(the
quantity that minimizes the total cost).
Total cost is based on:
Ordering cost = Co
STECO estimates the cost of placing an order
for NCs, regardless of the number of units
ordered, to be
$20 (clerical & purchasing agent labor)
+ $ 5 (material & telecommunications costs)
$25
Holding cost = Ch
STECO estimates the cost of holding an NC in
inventory for one year is
20% (opportunity cost)
+ 4% (variable cost)
24% of its purchase price. Since each NC
costs $8, the holding cost = $8 x .24 = $1.92
First derive an expression for the annual _______
and ordering cost (AHO) as a function of the
order__________.
The Annual Ordering Cost:
Annual Ordering Cost = Co x (no. of orders/yr)
Total Demand = 60,000
Order Quantity = 5,000
Therefore, STECO will place
60,000/5,000 = 12 orders per year
The general formula is N = D/Q
where N = no. orders/yr
D = annual demand
Q = order quantity
Annual Ordering Cost = CoN = Co(D/Q)
The Annual Holding Cost:
1. Annual _________cost is equal to Ch times
the average inventory
2. The average ________is equal to ½ of the
maximum inventory when demand occurs
at a __________rate.
Annual Holding Cost = Ch(Q/2)
Therefore, add the two expressions to get:
AHO(Q) = Co (D/Q) + Ch(Q/2)
Now, substituting the values, we get
AHO(Q) =$25(60,000/Q) + 1.92(Q/2)
= (1,500,000/Q) + 0.96Q
When Q = 5,000 it is seen that
AHO(5,000) = $300 + $4,800 = $5,100
This graph shows the optimal order quantity that
minimizes AHO(Q):
Dollars
6,000
AHO(Q) = Annual holding and ordering cost
5,000
4,000
Annual holding cost = 0.96Q
3,000
2,000
1,000
Annual ordering cost = (1,500,000)/Q
0
1,000
2,000
3,000
4,000
5,000
Here is the spreadsheet model for this example:
This model is
_______because
of the Q in the
denominator of
the ordering cost
formula.
Here are the solver parameters:
This ___________
(specifying that
there be at least
one order per
year) is included
to prevent Solver
from testing an
unreasonable
____________.
The EOQ Formula: Q*
Q* is the optimal order quantity or __________
order quantity expressed in terms of Co, Ch and D.
To develop a _____________expression for Q*,
first set annual holding cost equal to annual
ordering cost.
Ch(Q*/2) = Co (D/Q*)
Solving for Q* we get
(Q*)2 = 2CoD/Ch
Q* = 2CoD/Ch
Ch can be estimated with i [a ___________of the
purchase price (P)] and P, this equation can be
rewritten as:
Q* = 2C D/iP
o
Remember, P = $8, i (the fraction of P that is
used to calculate Ch) is 0.24 and D is 60,000.
Now, solve for Q*
Q* = 2(60,000)25/1.92
Q* = 1250
Now that we have Q*, the optimal order
quantity, find the AHO(Q*):
AHO(Q*) =AHO(1250)
= (1,500,000)/1250 + (0.96)(1250)
= $1200 + $1200
= $2400
Sensitivity Analysis
Now we ask: How ________are the results of the
model to the assumptions and the data?
STECO should be concerned about how sensitive
the ______________________and the optimal
annual cost are to the data.
If STECO errs in estimating the _________Co and
Ch, how much effect will that error have on the
difference between the __________Q* and AHO*
and the true Q* and AHO*.
If the results are highly sensitive to the values of
the estimates, should the optimal policy for the
model be________________?
To see how the ________results will vary with
changes in the holding and ___________cost
estimates, consider four cases in which the true
___________are different from the values
selected by STECO:
In STECO’s case, the EOQ model is __________to
approximately 10% variations in cost estimates.
Inventory with Quantity
Discounts Model
The following examples are variations in the
“classic” EOQ model.
Quantity Discounts
and STECO’s Overall Optimum:
Previously, the cost of purchasing the product
was assumed to be a________, independent of Q.
However, STECO’s NC supplier will offer a
quantity ____________as an incentive for more
business.
The supplier has agreed to offer a $0.10 discount
on every NC purchased if STECO orders in lots of
at least 5000 items.
Higher order quantities will ______the number of
orders placed and increase the average inventory
level, resulting in a higher annual ________cost.
The question is, will the discount be
advantageous to STECO? To answer this, first
develop an ____________curve and then find the
order quantity that minimizes it.
Let
ATC(Q) be the annual total cost
AHO(Q) be the sum of the annual holding
and ordering cost
APC be the annual purchase cost
ATC(Q) = AHO(Q) + APC
{
AHO(Q) = Co(D/Q) + iP(Q/2)
APC = PD
Ch
ATC(Q) = Co(D/Q) + iP(Q/2) + PD
Let P = $8.00 per unit, the Regular price
equation is:
ATC(Q) = 25(60,000)/Q + .24(8.00)(Q/2) + 8.00(60,000)
Let P = $7.90 per unit, the discount price
equation is:
ATC(Q) = 25(60,000)/Q + .24(7.90)(Q/2) + 7.90(60,000)
The general shapes of the Regular and Discount
curves are shown below:
ATC(Q)
Regular Price
Note that the
discount
curve lies
below the
regular cost
curve.
Discount Price
Q*R Q*D
Q
Also note that the value of Q, say Q*D, that
___________the discount price ATC(Q) is larger
than the value of Q, say Q*R, that minimizes the
________price = ATC(Q).
Now, assuming that the __________price holds
only if STECO orders at least B items at a time.
Two situations could arise:
The dark line portions indicate the ____________
function that STECO faces.
ATC(Q)
ATC(Q)
Regular Price
Regular Price
Discount
Price
Discount
Price
Q*R Q*D
B
Q
Q*R Q*D
Q
B
If B < Q*D, STECO will achieve the __________
cost by ordering Q*D.
If B > Q*D, the optimal decision is not obvious.
The general rule is:
If B < Q*D,
If B > Q*D,
order
order
Q*D
Q*R if regular price <
discount price
ATC(Q*R) < ATC(B)
B
if not
Here is the Excel version of the quantity discount
inventory model:
Solver optimizes a ______integer nonlinear program
(MINLP) to evaluate the two EOQ functions, one with
and without the_____________.
Here are the formulas:
Inventory and Production,
A Lot Size Model
STECO has an extensive and modern heattreatment fibre cable “_________” facility that it
uses to produce a number of specialty cable items
that it then holds in______________.
Two important characteristics of this facility:
1. There is a large _____cost associated with
producing each cable product
2. Once the setup is complete, production is
at a _______and known rate.
Setup cost (____________to ordering cost) is
incurred because it is necessary to change the
plastic fibre molds and the operating temperature
in the heat-treatment facility to meet the
______________set forth by the cable standards
specification.
Each cable must have ___________attached and
under go testing for frequency response.
An order quantity of network cables arrives from
production into inventory steadily over a period
of several days.
A modification in the EOQ formula is required.
Consider a product in which
d = no. of units demanded each day
p = no. of units produced each day during a
production run
Co = setup cost that is independent of the
quantity produced
ch = cost per day of holding inventory
Note that if p < d, demand is greater than
STECO’s ability to produce.
Below is a plot of inventory on hand for the
Production Lot Size model:
Inventory
On hand
Rate of
decrease
Rate of
increase
d
p-d
Q
p
Q
d
Production
run
Time (Days)
Cycle time
The formulas are:
Max. inventory = (p-d)(Q/p)
Avg. inventory = ½ (p-d)(Q/p)
= Q/2(1-(p/d))
Holding Cost per Day = ch(Q/2) (1-(p/d))
Setup Cost per Day = Co/(Q/d) = Co(d/Q)
DHS(Q) = Co(d/Q) + ch(Q/2) (1-(p/d))
The value of Q that minimizes DHS(Q) for the
production lot size model is:
Q* =
2Cod
ch(1-(d/p))
Substituting Q* for Q in the expression for
DHS(Q) gives us an expression for the _________
daily holding and setup dost:
DHS(Q*) =
d
2Codch 1- 1p
Note that this expression does not _______on Q.
STECO must first estimate the various _________
and then obtain Q*. For illustration, let
Demand average 200 NCs per day
Setup cost is $100
Production rate is $400 NCs per day
Production cost is $1
Annual interest rate is 0.24
Number of working days per year is 240
Holding cost per day is
($1)(0.24)/240 = $0.001
The optimal production lot size for this product is:
Q* =
2(200)(100)
0.001(1-(200/400))
= 8944
The minimum daily holding and setup cost is:
= 2(200)(100)(0.001(1-(200/400)) = $4.47
A production run of this size yields a supply of
NCs large enough to satisfy demand for
8944 = 44.72 days
200
Here is the Excel model for this problem:
Here are the spreadsheet formulas: