Lecture
5
MGMT 650
Inventory Models – Chapter 11
1
Announcements

HW #3 solutions and grades posted in BB
 HW #3 average = 134.4 (out of 150)
 Final exam next week
 Open book, open notes….
 Final preparation guide posted in BB
 Proposed class structure for next week
Lecture – 6:00 – 7:50
 Class evaluations – 7:50 – 8:00
 Break – 8:00 – 8:30
 Final – 8:30 – 9:45

2
Inventory Management – In-class Example


Number 2 pencils at the campus book-store are sold at a fairly
steady rate of 60 per week. It cost the bookstore $12 to initiate an
order to its supplier and holding costs are $0.005 per pencil per
year.
Determine
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The optimal number of pencils for the bookstore to purchase to minimize total
annual inventory cost,
Number of orders per year,
The length of each order cycle,
Annual holding cost,
Annual ordering cost, and
Total annual inventory cost.
If the order lead time is 4 months, determine the reorder point.
Illustrate the inventory profile graphically.
What additional cost would the book-store incur if it orders in
batches of 1000?
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Management Scientist Solutions
4
Management Scientist Solutions
Chapter 11 Problem #4
EOQ
(Time between placing 2 consecutive orders - in days)
5
EOQ with Quantity Discounts

The EOQ with quantity discounts model is
applicable where a supplier offers a lower
purchase cost when an item is ordered in larger
quantities.
 This model's variable costs are

annual holding,
 Ordering cost, and
 purchase costs.

For the optimal order quantity, the annual holding
and ordering costs are not necessarily equal.
6
EOQ with Quantity Discounts

Assumptions

Demand occurs at a constant rate of D items/year.
 Ordering Cost is $Co per order.
 Holding Cost is $Ch = $CiI per item in inventory
per year


Purchase Cost (C)



note holding cost is based on the cost of the item, Ci
$C1 per item if quantity ordered is between 0 and x
$C2 if order quantity is between x1 and x2 , etc.
Lead time is constant
7
EOQ with Quantity Discounts

Formulae

Optimal order quantity: the procedure for
determining Q * will be demonstrated
 Number of orders per year: D/Q *
 Time between orders (cycle time): Q */D years
 Total annual cost: (formula 11.28 of book)
Q*
D
TC 
Ch 
C0  D.C
*
2
Q
(holding + ordering + purchase)
8
Example – EOQ with Quantity Discount
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Walgreens carries Fuji 400X instant print film
The film normally costs Walgreens $3.20 per roll
Walgreens sells each roll for $5.25
Walgreens's average sales are 21 rolls per week
Walgreens’s annual inventory holding cost rate is 25%
It costs Walgreens $20 to place an order with Fujifilm, USA
Fujifilm offers the following discount scheme to Walgreens
 7% discount on orders of 400 rolls or more
 10% discount for 900 rolls or more, and
 15% discount for 2000 rolls or more
Determine Walgreen’s optimal order quantity
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Management Scientist Solutions
10
Economic Production Quantity (EPQ)

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The economic production quantity model is a variant
of basic EOQ model
Production done in batches or lots
A replenishment order is not received in one lump sum
unlike basic EOQ model
Inventory is replenished gradually as the order is
produced

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This model's variable costs are



hence requires the production rate to be greater than the
demand rate
annual holding cost, and
annual set-up cost (equivalent to ordering cost).
For the optimal lot size,

annual holding and set-up costs are equal.
11
EPQ = EOQ with Incremental Inventory
Replenishment
12
EPQ Model Assumptions
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Demand occurs at a constant rate of D items per year.
Production rate is P items per year (and P > D ).
Set-up cost: $Co per run.
Holding cost: $Ch per item in inventory per year.
Purchase cost per unit is constant (no quantity
discount).
Set-up time (lead time) is constant.
Planned shortages are not permitted.
13
EPQ Model Formulae

Optimal production lot-size (formula 11.16 of book)
Q*=
2DCo /[(1-D/P )Ch]

Number of production runs per year: D/Q *

Time between set-ups (cycle time): Q */D years

Total annual cost (formula 11.14 of book)

[(1/2)(1-D/P )Q *Ch] + [DCo/Q *]
(holding + ordering)
14
Example: Non-Slip Tile Co.
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Non-Slip Tile Company (NST) has been using production runs of
100,000 tiles, 10 times per year to meet the demand of 1,000,000
tile annually.
The set-up cost is $5,000 per run
Holding cost is estimated at 10% of the manufacturing cost of $1
per tile.
The production capacity of the machine is 500,000 tiles per month.
The factor is open 365 days per year.
Determine

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
Optimal production lot size
Annual holding and setup costs
Number of setups per year
Loss/profit that NST is incurring annually by using their present
production schedule
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Management Scientist Solutions
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

Optimal TC = $28,868
Current TC = .04167(100,000) + 5,000,000,000/100,000
= $54,167
LOSS
= 54,167 - 28,868 = $25,299
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Lecture
5
Forecasting
Chapter 16
17
Forecasting - Topics

Quantitative Approaches to Forecasting
 The Components of a Time Series
 Measures of Forecast Accuracy
 Using Smoothing Methods in Forecasting
 Using Trend Projection in Forecasting
18
Time Series Forecasts

Trend - long-term movement in data
 Seasonality - short-term regular variations in
data
 Cycle – wavelike variations of more than one
year’s duration
 Irregular variations - caused by unusual
circumstances
 Random variations - caused by chance
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Forecast Variations
Irregular
variatio
n
Trend
Cycles
90
89
88
Seasonal variations
20
Smoothing Methods

In cases in which the time series is fairly stable
and has no significant trend, seasonal, or cyclical
effects, one can use smoothing methods to average
out the irregular components of the time series.
 Four common smoothing methods are:

Moving averages
 Weighted moving averages
 Exponential smoothing
21
Example of Moving Average


Sales of gasoline for the past 12 weeks at your local
Chevron (in ‘000 gallons). If the dealer uses a 3period moving average to forecast sales, what is the
forecast for Week 13?
Past Sales
Week
1
2
3
4
5
6
Sales
17
21
19
23
18
16
Week
7
8
9
10
11
12
Sales
20
18
22
20
15
22
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Management Scientist Solutions
MA(3) for period 4
= (17+21+19)/3 = 19
Forecast error for period 3
= Actual – Forecast =
23 – 19 = 4
23
MA(5) versus MA(3)
Actual
1
2
3
4
5
6
7
8
9
10
11
12
MA(3)
17
21
19
23
18
16
20
18
22
20
15
22
MA(5)
MA Forecast Graph
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21
20
19
18
18
20
20
19
19.6
19.4
19.2
19
18.8
19.2
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Actual/MA Forecast sale
values
Week
25
20
Actual
15
MA(3)
10
MA(5)
5
0
1
2 3
4
5
6 7
8
9 10 11 12
Week
24
Exponential Smoothing
• Premise - The most recent observations might have the highest
predictive value.

Therefore, we should give more weight to the more recent time periods
when forecasting.
Ft+1 = Ft + (At - Ft)
25
Linear Trend Equation
Suitable for time series data that exhibit a long term linear trend
Ft
Ft = a + bt
a
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0 1 2
Ft = Forecast for period t
t = Specified number of time periods
a = Value of Ft at t = 0
b = Slope of the line
3 4 5
t
26
Linear Trend Example
Linear trend equation
F11 = 20.4 + 1.1(11) = 32.5
Sale increases every time
period @ 1.1 units
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Actual vs Forecast
Actual/Forecasted sales
Linear Trend Example
35
30
25
20
Actual
15
Forecast
10
5
0
1
2
3
4
5
6
7
8
9
10
Week
F(t) = 20.4 + 1.1t
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Measure of Forecast Accuracy

MSE = Mean Squared Error
Week # Actual (A) Forecast(F) Error =E =A-F E(squared)
1
21.6
21.5
0.1
0.01
2
22.9
22.6
0.3
0.09
3
25.5
23.7
1.8
3.24
4
21.9
24.8
-2.9
8.41
5
23.9
25.9
-2
4
6
27.5
27
0.5
0.25
7
31.5
28.1
3.4
11.56
8
29.7
29.2
0.5
0.25
9
28.6
30.3
-1.7
2.89
10
31.4
31.4
0
0
Sum of E(squared)
30.7
MSE=
3.07
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Forecasting with Trends and Seasonal
Components – An Example
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Business at Terry's Tie Shop can be viewed as falling into three
distinct seasons: (1) Christmas (November-December); (2) Father's
Day (late May - mid-June); and (3) all other times.
Average weekly sales ($) during each of the three seasons
during the past four years are known and given below.
Determine a forecast for the average weekly sales in year 5 for each
of the three seasons.
Year
Season
1
2
3
4
1
1856 1995 2241 2280
2
2012 2168 2306 2408
3
985 1072 1105 1120
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Management Scientist Solutions
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Interpretation of Seasonal Indices

Seasonal index for season 2 (Father’s Day) = 1.236


Means that the sale value of ties during season 2 is 23.6%
higher than the average sale value over the year
Seasonal index for season 3 (all other times) = 0.586

Means that the sale value of ties during season 3 is 41.4%
lower than the average sale value over the year
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Lecture
5
Decision Analysis
Chapter 14
33
Decision Environments

Certainty - Environment in which relevant
parameters have known values

Risk - Environment in which certain future
events have probabilistic outcomes

Uncertainty - Environment in which it is
impossible to assess the likelihood of
various future events
34
Decision Making under Uncertainty
Maximin - Choose the alternative with
the best of the worst possible payoffs
Maximax - Choose the alternative with
the best possible payoff
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Payoff Table: An Example
Possible Future Demand
Small
facility
Medium
facility
Large
facility
Low
Moderate
High
$10
$10
$10
7
12
12
-4
2
16
Values represent payoffs (profits)
36
Maximax Solution
Note: choose the
“minimize the
payoff” option if
the numbers in
the previous slide
represent costs
37
Maximin Solution
38
Minimax Regret Solution
39
Decision Making Under Risk - Decision Trees
Payoff 1
Decision Point
Chance Event
Payoff 2
2
Payoff 3
1
B
Payoff 4
2
Payoff 5
Payoff 6
40
Decision Making with Probabilities

Expected Value Approach

Useful if probabilistic information regarding the
states of nature is available
 Expected return for each decision is calculated by
summing the products of the payoff under each
state of nature and the probability of the
respective state of nature occurring
 Decision yielding the best expected return is
chosen.
41
Example: Burger Prince

Burger Prince Restaurant is considering opening a new restaurant
on Main Street.

It has three different models, each with a different seating
capacity.

Burger Prince estimates that the average number of customers per
hour will be 80, 100, or 120 with a probability of 0.4, 0.2, and 0.4
respectively

The payoff (profit) table for the three models is as follows.
•
s1 = 80 s2 = 100 s3 = 120
Model A
$10,000 $15,000
$14,000
Model B
$ 8,000 $18,000
$12,000
Model C
$ 6,000 $16,000
$21,000
Choose the alternative that maximizes expected payoff
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Decision Tree
d1
1
d2
d3
2
3
4
s1
s2
s3
.4
.2
.4
s1
.4
s2
s3
.2
s1
s2
s3
.4
.4
.2
.4
Payoffs
10,000
15,000
14,000
8,000
18,000
12,000
6,000
16,000
21,000
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Management Scientist Solutions
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