Lecture

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Lecture5

MGMT 650Inventory Models – Chapter 11

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AnnouncementsAnnouncements 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

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Inventory Management – In-class ExampleInventory 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?

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Management Scientist SolutionsManagement Scientist Solutions

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Management Scientist Solutions Management Scientist Solutions Chapter 11 Problem #4Chapter 11 Problem #4

EOQ

(Time between placing 2 consecutive orders - in days)

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EOQ with Quantity DiscountsEOQ 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.

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EOQ with Quantity DiscountsEOQ 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

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EOQ with Quantity DiscountsEOQ 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)

(holding + ordering + purchase)

CDCQ

DC

QTC h .

2 0*

*

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Example – EOQ with Quantity DiscountExample – 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 more7% discount on orders of 400 rolls or more 10% discount for 900 rolls or more, and10% discount for 900 rolls or more, and 15% discount for 2000 rolls or more15% discount for 2000 rolls or more

Determine Walgreen’s optimal order quantityDetermine Walgreen’s optimal order quantity

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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 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.

Economic Production Quantity (EPQ)Economic Production Quantity (EPQ)

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EPQ = EOQ with Incremental Inventory EPQ = EOQ with Incremental Inventory ReplenishmentReplenishment

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EPQ Model AssumptionsEPQ 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.

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EPQ Model FormulaeEPQ 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)

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Example: Non-Slip Tile Co.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

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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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Lecture5

ForecastingChapter 16

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Forecasting - TopicsForecasting - 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

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Time Series ForecastsTime 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 VariationsForecast Variations

Trend

Irregularvariation

Seasonal variations

908988

Cycles

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Smoothing MethodsSmoothing 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

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

Example of Moving Average

Past Sales

WeekWeek SalesSales WeekWeek SalesSales 1 17 7 201 17 7 20 2 21 8 182 21 8 18 3 19 9 223 19 9 22 4 23 10 204 23 10 20 5 18 11 155 18 11 15

6 166 16 12 12 22 22

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MA(3) for period 4

= (17+21+19)/3 = 19

Forecast error for period 3 = Actual – Forecast = 23 – 19

= 4

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MA(5) versus MA(3)MA(5) versus MA(3)

Week Actual MA(3) MA(5)1 172 213 194 23 195 18 216 16 20 19.67 20 19 19.48 18 18 19.29 22 18 19

10 20 20 18.811 15 20 19.212 22 19 19

MA Forecast Graph

0

5

10

15

20

25

1 2 3 4 5 6 7 8 9 10 11 12

Week

Actu

al/M

A Fo

reca

st s

ale

valu

es

Actual

MA(3)

MA(5)

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Exponential SmoothingExponential 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)

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Linear Trend EquationLinear Trend Equation

Ft = Forecast for period t t = Specified number of time periods a = Value of Ft at t = 0 b = Slope of the line

Ft = a + bt

0 1 2 3 4 5 t

Ft

a

Suitable for time series data that exhibit a long term linear trend

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Linear Trend ExampleLinear Trend Example

F11 = 20.4 + 1.1(11) = 32.5

Linear trend equation

Sale increases every time period @ 1.1 units

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Actual vs ForecastActual vs Forecast

Linear Trend Example

0

5

10

15

20

25

30

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1 2 3 4 5 6 7 8 9 10

Week

Act

ual

/Fo

reca

sted

sal

es

Actual

Forecast

F(t) = 20.4 + 1.1t

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Measure of Forecast AccuracyMeasure 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 0.012 22.9 22.6 0.3 0.093 25.5 23.7 1.8 3.244 21.9 24.8 -2.9 8.415 23.9 25.9 -2 46 27.5 27 0.5 0.257 31.5 28.1 3.4 11.568 29.7 29.2 0.5 0.259 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 Forecasting with Trends and Seasonal Components – An ExampleComponents – 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 seasonsduring 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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Interpretation of Seasonal IndicesInterpretation 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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Lecture5

Decision AnalysisChapter 14

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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

Decision EnvironmentsDecision Environments

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Maximin - Choose the alternative with the best of the worst possible payoffs

Maximax - Choose the alternative with the best possible payoff

Decision Making under UncertaintyDecision Making under Uncertainty

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Payoff Table: An ExamplePayoff Table: An Example

Low Moderate High

Small facility

$10 $10 $10

Medium facility

7 12 12

Large facility

- 4 2 16

Possible Future Demand

Values represent payoffs (profits)

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Maximax SolutionMaximax Solution

Note: choose the “minimize the payoff” option if the numbers in the previous slide represent

costs

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Maximin Solution Maximin Solution

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Minimax Regret SolutionMinimax Regret Solution

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Decision Making Under Risk - Decision Decision Making Under Risk - Decision TreesTrees

State of nature 1

B

Payoff 1

State of nature 2

Payoff 2

Payoff 3

2

Choose A’1

Choose A’2

Payoff 6State of nature 2

2

Payoff 4

Payoff 5

Choose A’3

Choose A’4

State of nature 1

Choose A

’

Choose A’2

1

Decision PointChance Event

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Decision Making with ProbabilitiesDecision 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.

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Example: Burger PrinceExample: 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 TreeDecision Tree

1111

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.4.4

.2.2

.4.4

dd11

dd22

dd33

ss11

ss11

ss11

ss22

ss33

ss22

ss22

ss33

ss33

PayoffsPayoffs

10,00010,000

15,00015,000

14,00014,0008,0008,000

18,00018,000

12,00012,000

6,0006,000

16,00016,000

21,00021,000

2222

3333

4444

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