1. Inventory Models – Chapter 11
Lecture 5
MGMT 650
Inventory Models – Chapter 11

2. Announcements HW #3 solutions and grades posted in BB
HW #3 average = (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

3. 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
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?

4. Management Scientist Solutions

5. Management Scientist Solutions Chapter 11 Problem #4
EOQ
(Time between placing 2 consecutive orders - in days)

6. 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.

7. 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
note holding cost is based on the cost of the item, Ci
Purchase Cost (C)
$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

8. 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 of book)
(holding + ordering + purchase)

9. Example – EOQ with Quantity Discount
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

10. Management Scientist Solutions

11. Economic Production Quantity (EPQ)
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
hence requires the production rate to be greater than the demand rate
This model's variable costs are
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.

12. EPQ = EOQ with Incremental Inventory Replenishment

13. EPQ Model Assumptions
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.

14. 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 of book)
[(1/2)(1-D/P )Q *Ch] + [DCo/Q *]
(holding + ordering)

15. Example: Non-Slip Tile Co.
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
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

16. Management Scientist Solutions
Optimal TC = $28,868
Current TC = (100,000) + 5,000,000,000/100,000
= $54,167
LOSS = 54, ,868 = $25,299

17. Lecture 5
Forecasting
Chapter 16

18. 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

19. 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

20. Forecast Variations Trend Cycles Irregular variation 90 89 88
Seasonal variations

21. 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

22. Example of Moving Average
Sales of gasoline for the past 12 weeks at your local Chevron (in ‘000 gallons). If the dealer uses a 3-period moving average to forecast sales, what is the forecast for Week 13?
Past Sales
Week Sales Week Sales

23. Management Scientist Solutions
MA(3) for period 4
= ( )/3 = 19
Forecast error for period 3 = Actual – Forecast = 23 – 19 = 4

24. MA(5) versus MA(3)

25. 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)

26. Suitable for time series data that exhibit a long term linear trend
Linear Trend Equation
Suitable for time series data that exhibit a long term linear trend Ft
Ft = a + bt a
Ft = Forecast for period t
t = Specified number of time periods
a = Value of Ft at t = 0
b = Slope of the line t

27. Sale increases every time period @ 1.1 units
Linear Trend Example
Linear trend equation
F11 = (11) = 32.5
Sale increases every time 1.1 units

28. Actual/Forecasted sales
Actual vs Forecast
Linear Trend Example 35 30 25
Actual/Forecasted sales 20
Actual 15
Forecast 10 5 1 2 3 4 5 6 7 8 9 10
Week
F(t) = t

29. Measure of Forecast Accuracy
MSE = Mean Squared Error

30. Forecasting with Trends and Seasonal Components – An Example
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

31. Management Scientist Solutions

32. 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

33. Lecture 5
Decision Analysis
Chapter 14

34. 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

35. 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

36. Payoff Table: An Example
Possible Future Demand
Low
Moderate
High
Small facility
$10
Medium facility 7 12
Large facility
- 4 2 16
Values represent payoffs (profits)

37. Maximax Solution
Note: choose the “minimize the payoff” option if the numbers in the previous slide represent costs

38. Maximin Solution

39. Minimax Regret Solution

40. Decision Making Under Risk - Decision Trees
State of nature 1 B
Payoff 1
State of nature 2
Payoff 2
Payoff 3 2
Choose A’1
Choose A’2
Payoff 6
Payoff 4
Payoff 5
Choose A’3
Choose A’4
Choose A’ 1
Decision Point
Chance Event

41. 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.

42. 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 = s2 = s3 = 120
Model A $10, $15, $14,000
Model B $ 8, $18, $12,000
Model C $ 6, $16, $21,000
Choose the alternative that maximizes expected payoff

43. Decision Tree Payoffs 10,000 2 15,000 d1 14,000 8,000 d2 1 3 18,000 d3.4
10,000 s2 .2 2
15,000 s3 .4 d1
14,000 .4 s1
8,000 d2 1 3 s2 .2
18,000 d3 s3 .4
12,000 s1 .4
6,000 4 s2 .2
16,000 s3 .4
21,000

44. Management Scientist Solutions