MGTSC 352

Finals Notes

Forecasting

Forecasting – Quantitative

• Time series analysis: uses only past records of demand to forecast future demand – moving averages – exponential smoothing– ARIMA

• Causal methods: uses explanatory variables (timing of advertising campaigns, price changes)– multiple regression– econometric models

Choosing a Forecasting Method

START

Are accurate historical data available?

Is forecast important?

Is forecast important?

Flip a coin; use your intuition;

look at your horoscope; consult an economist

Are you willing to pay for greater

accuracy?

Is there at least 1-2 months before

forecast is needed?

Select appropriate qualitative method

Use a causal method

Use time-series method

END

Yes

No Yes No Yes

No

No

Yes Yes No

Simple models

• Notation– Dt = Actual demand in time period t

– Ft = Forecast for period t

– Et = Dt - Ft = Forecast error for period t

Simple models

• Last Point model– Yesterday’s demand will be today’s forecast

• Ft+1 = Dt

– Not a good model cause it does no learning, it is just a follower

• Average Model– Take the average of all known data to make

prediction– Ft+1 = average(D1,D2, . . . , Dt)– Very old data is probably useless to us, so why

include in the model?

Simple models

• Simple moving average (SMA)– Take the average of data in the past ‘m’ periods as a forecast– Ft+1= average(Dt-m+1, Dt-m+2, . . . , Dt

– Maybe some data points should have more weight on them than others

– Can’t begin to forecast until you have the specified window of data points

• Weighted moving average– Like SMA but different points have different levels of importance,

or weight– Ft+1 = W1*Dt-m+1+W2*Dt-m+2+ . . . + Wm*Dt

– ΣWi = 1– How to choose the weights?– Data that is closer to the forecast point should have more weight

placed on it

Simple Exponential Smoothing (SES)

• Initialization: F2 = D1

• Learning: Ft+1 = LS*Dt + (1-LS)*Ft

• Predicting: Ft+k = Ft+1

– Future forecasts are equal to the last forecast

• LS = level smoother– Lower LS, higher the volatility– Higher LS, lower the volatility, however we will be doing less

prediction and more following• Must have some sort of compromise

– At LS=0, all values will be same as the initialized forecast,– At LS=1, the graph will follow the demand perfectly one day late

» Really worthless

Performance Measures

•Remember that for performance measures, you must have a demand and a corresponding forecast. •What is error if there is nothing to compare to?

Bias

• Average error

• (1/n)Σ Et

• Positive and negative values will cancel out– May appear to have a small bias but large

errors in the positive and negative direction will cancel each other out

Mean Absolute Deviation (MAD)

• (1/n) Σ |Et|

• Similar to bias, but no negative values

• No way of telling which direction the error is occurring, just that there is error

Mean Square Error (MSE) andStandard Error (SE)

• (1/(n-1)) Σ Et2

• Places more emphasis on large error terms• Heavily influenced by outlier terms and

also not in units of the error

• SE = (MSE)1/2

• Same units as error• Dampens the effect of the error terms

Mean Absolute Percentage Error (MAPE)

• (1/n) Σ (Et/Dt)

• In terms of percentage, which is good

• If error is large but Dt is humongous, MAPE will appear to be small– That is the problem when you have you

performance measure influenced by the demand

More Complex Forecasting Models

Double Exponential Smoothing (DES)

• Takes level and trend into account• Trend = change in the level• Initialization:

– L2 = (D1+D2)/2– T2 = D2-D1

• Learning:– Lt = LS*Dt+(1-LS)(Lt-1+Tt-1)– Tt = TS(Lt-Lt-1)+(1-TS)Tt-1

DES

• Prediction:– Ft+1 = Lt+Tt

– Predicting k steps into the future• Ft+k = Lt+kTt

• Change is exaggerated at beginning• As time moves on, the emphasis a single data

point has on the model decreases (less fluctuation)

• Remember that you can only make level and trend up to data points that you have, can’t make level and trend for predicted data

Triple Exponential Smoothing (TES)

• Takes level, trend and seasonality into account

• Use if you see seasonal trends in the data and if the data appears to be non-linear

TES - Initialization

• P is the number of seasons in a dataset– Ex) weeky = 7, monthly = 12, hourly = 24,

quarterly = 4

• The initialization window is equal to P, we allow P terms to pass before we begin to forecast

TES - Initialization

• Level: average of initial ‘P’ data points = A• Trend: Tp = (Dp+1-D1)/P

– Ex) if we have monthly data from Jan 2000 to Dec 2005, initialization would be:

(DJan2001-DJan2000)/12• Seasonality: Si = Di/A

– The initial seasonality indices are ratios of that period’s demand against the average of the initial ‘P’ data points (the level)Notice that whenever we deal with seasonality, we always match up corresponding seasons with one another. January matches up with January, Tuesday matches up with Tuesday, etc.

TES - Learning

• The one-time forecast is

Ft+1 = (Lt+Tt)/St+1-P

• If Ft+1 is a Tuesday, St+1-P is the previous Tuesday’s seasonality

TES - Learning

• Lt = LS(Dt/St-p)+(1-LS)(Lt-1+Tt-1)

• We want to remove the effect of seasonality from level

• We want to know what December sales would be like if it weren’t Christmas, we want the deseasonalized level

• The old data is already deseasonalized(Lt-1+Tt-1)(St+1-p/St+1-p)

TES - Learning

• Tt = TS(Lt-Lt-1)+(1-TS)Tt-1

– Trend is not effected by seasonality, it is merely a change in level

– Same as in DES

• St = SS(Dt/Lt)+(1-SS)St-p

– Don’t want the effects of level to effect seasonality, so we divide it out

– Always remember to match up seasons with the same season

TES - Predicting

Ft+k = (Lt+kTt)St+k-P k<=p• We can only use the last level and trend

for the prediction• For seasonality we can only use the

seasonality for the last cycle.– When K>p, we must reset seasonality to the

beginning of the cycle– We can not create seasonality for data we

don’t have, just like level and trend

Solver

• When using solver to optimize forecasting problems, minimize MSE or SE cause they are linear functions– If using bias, make it so that bias = 0

• Solver is used to optimize the smoothing factors so that we can achieve the optimum performance measure

• Boundaries must ve set in solver so that we aren’t chasing randomness and so we are actually predicting data– Set bounds so that 0.05<smoothers<0.95

Simple Linear Regression with Seasonality Indices (SLRwSI)

• Use this method if you have seasonal data and you can see a linear trend in the dataset

• SLR is intended to make bias = 0

• SLR = Intercept + k*slope– Functions can be imputted by separate functions in

excel or by adding treadline, but will be optimized by solver anyways, so it doesn’t really make a difference

SLRwSI

• Seasonality is not updated in SLRwSI, we will have a different seasonal index for each period and the dataset will cycle through those initial indices

The average of the seasonal indices must equal 1

• SLRwSI = SLR*SI• To optimize

– Minimize performance measure– Allow manipulation of slope, intercept and all of the

SI’s– Restrain so that the average of the SI’s must equal 1

Limitation of Solver

• Solver tries to find the minimum value of the dataset, but that minimum may be the local minimum, we want the global minimum

• To help fix this, start the indices or the SI’s at different points to find the true minimum

• 95% Prediction Interval = Forecast ± 2*SE

Simulation

Simulation

• Simulating a value from a Normal Distribution:Breaking the formula down

• ROUND(NORMINV(RAND(),mean,stdev),0)• Step 1: generate random number

RAND()• Step 2: convert random number to normal distribution

NORMINV(RAND(),mean,stdev)• Step 3: round to whole number

ROUND(NORMINV(RAND(),mean,stdev),0)

Bard Outside Example: A “Newsvendor Problem”

• Bard Outside:– Decision: # of seats– Uncertain future

demand– Demand > # of seats

lost revenue– Demand < # of seats

empty seats

• A newsvendor:– Decision: # of

newspapers to get– Uncertain future

demand– Demand > # of papers

lost revenue– Demand < # of papers

disposal costs

Simulation notes

• To find % of something above x amount– 1-normdist(X,mean,stddev,true)

• To find simulated demand, use norminv function– Use max(0,round(...)), keeps it non-negative

• For table, highlight the variable cells and the count cells. – Highlight as many sells as the size of table that you

want

• DONT FORGET TO FREEZE TABLE AFTER MAKING

Distributions

• Triangular distributions– Have min, max, and most likely scenario

• Adding two random values will give triangular dist.

• Uniform distribution– If you added one random variable

• Trapezoidal distribution

Aggregate Planning

Aggregate Planning

• Solver error: Unbounded Problem

• How will you know: The set cell values do not converge

• What it means:– Possible to achieve infinite profit

• Either you will become filthy rich, or (more likely) there is something wrong with your model

• How to fix it: look for missing constraints

Aggregate Planning

• Solver Error: Infeasible Problem

• How will you know: Solver could not find a feasible solution

• What it means:– Impossible to satisfy all constraints

• Possible reasons:– You need more resources– You over-constrained the problem

Sensitivity Report

• Allowable increase/decrease– The bounds that we are allowed to be within for our answer to

still be the optimal one.– Once we go beyond these bounds we would have to re-solve the

answer

• Shadow price – How much the objective function will change if we change the

R.H side of the corresponding constraint within the limits given in the allowable increase/decrease columns

• How much the change in the target cell if the RH side of a constraint increases by 1

• If we have unconsumed materials, shadow price will always be 0

– Deals with marginal pricing

Sensitivity report

• Reduced cost– For each variable which is currently zero, an estimate

of how much the objective function will change if we force that variable to be non-zero.

– Think opportunity cost

• To decide whether the objective function will go up or down use: – constraint more (less) restrictive after change in right-

hand side implies objective function worse (better) – if objective is maximise (minimise) then worse means

down (up), better means up (down)

Tolerance

• We should set tolerance to as low as possible so that solver will find the absolute optimal solution, instead of a value within a certain tolerance interval.– May take a little more time to solve if we have

a low tolerance level

Level vs. Chase

• Level– Always producing at a constant rate no matter

what the demand is

• Chase– Attempting to chase the simulated demand

Distribution Planning

Distribution Planning

• What should overall distribution system be?• Where should inventories of products or raw

materials be stored?• How much inventory of each product and raw

material should be stored at each location• How should the flow of products and raw

materials through the distribution be coordinated• What models of transportation should be used?

Distribution Planning

• All distribution problems are really special case of minimum cost problem, even the shortest distance problem, which replaces distances with cost

• Remember to freeze cells when using the sumif function

• Hit ctrl + ~ to get into formula mode, will make it much easier to debug

Distribution Planning

if demand is greater than supply, solver will try to solve to satisfy the demand but there wont be enough supply, so it will cause an error (infeasible solution)

if you have a node with no demand or supply, flow will come in and won't stay cause there is nothing required and will immediately flow out, like intersection

Distribution Planning

• Shortest path problem– If we are required to go to a certain path, best way is

to solve it in two parts• 1st part is when we go from supply city to intermediate path• 2nd part is when we go from intermediate city to final demand

path

– Set demand = 1 at destination city and set supply = 1 at city of origin

– Make sure that supply + flowin = demand + flow out• This will allow us to make a path with no jumps

Distribution Planning

• Shortest Path problem cont.– If we have to traverse a specific arc, but not to a

specific city to within that arc, before going to a specific city, make sure you allow for two-way travel

• In three cells, have: – city 1 -> city 2– City 2 -> city 1– sum

• Each path will reference truckload along that path• Sum is the sum of the two arcs• Constrain solver so that the sum>=1, that way it must

traverse the path but also allows for back travel

Distribution Planning

• New locations• If wondering whether or not to open a new

facility, use a binary variable• To ensure that we don’t produce if we don’t

open:– Set an upper bound = max prod * binary– Constrain solver so that production can not be greater

than the upper bound

• Must constrain solver so that supply + flow in + production >= demand + flow out

Inventory Management

Inventory Management

• Goods that have not yet been sold• Keep inventory when

– Demand unpredictable– Delivery takes time– Fixed cost for delivery

• Relevant question– When to order (ROP = Reorder point)– How much to order (Q = reorder quantity)

• MAKE SURE TIME UNITS ARE CONSISTENT, DON’T MIX YEARS WITH MONTHS

Relevant Costs

• Acquisition cost ($/unit purchased)

• Ordering costs($/order)– clerical expenses– delivery, inspection– setup (prod.)

• Carrying costs = Holding costs($/unit/time unit)– cost of capital– insurance– shrinkage, spoilage, obsolescence– material handling (fork lifts, space)

• Shortage costs($/unit short)– lost goodwill, discounts, penalties– lost sales– shut down of assembly line (prod.)

Time

Inventory

ROP

Leadtime

Q

Maximum inventory

Minimum inventory

Avg. inventory

LTD = Demand

during leadtime

Histogram

Need 3 columns– # sold, bins, and frequency as headers– # sold will be a range (0-2, 3-4, etc..)– Bins refers to values at or below that value

• 2 means 0-2, 4 means 3-4, etc..

– Frequency means how often value corresponding to a bin shows up in the dataset

Histogram

• Highlight the empty frequency cells

• Type in frequency (data, bins)

• While they are all highlighted, hit ctrl + shift + enter, this will cause the frequency to appear in the cells

Histogram

• in the graph template, hit column graph

• Highlight # sales and frequency to be graphed

• Once graph is made, double click the graph and under options you can change the distance between the columns to be 0

• Histogram complete

Simulation

• Orders take time to come into your place of business, this time will affect how your business is run because it will effect your reorder points and order quantities

• Beginning inventory is equal to ending inventory of the previous day + the order that came in that day

Simulation

• Inventory position is beginning inventory plus inventory that is in transit. If we ordered two day ago, and we know we will get the inventory in 5 days, then we wouldn’t order more stock because we know that we have an order on the way

• If a new order just arrives and it is too short, then we would put a new order through

Simulation

• Order if demand is greater than the inventory position– We can use if statements to ensure this– If(demand>=inventory position, order amount

Q, else don’t place order)

• Sales will be the minimum of demand or beginning inventory, not inventory position because that inventory is not in the store– Min(demand, beginning inventory)

Simulation

• Shortage is demand less sales

• Ending inventory is beginning inventory less sales. If your order will come in at the end of the business day, then ending inventory will include this as well

• Holding cost is the average of beginning and ending inventory, multiplied by the holding cost per unit

Simulation

• Fill rate is the amount of demand that is satisfied by the inventory

• =total sales/total demand

Tables in Excel

• Say we want to see how net profit varies with differing ROP and Q

• Put values for ROP and Q along row and column, except leave the top left corner of the table blank

• In the top left corner, reference net profit• Highlight entire area, then go data->table• For row, reference original value for row, and for

column, reference original value for column• Can use conditional formatting to highlight the max

amount or to highlight minimum amounts, if you require that we must reach a certain profit, or fill rate or whatever

EOQ = Economic Order Quantity

• Assumptions– Demand is constant– Inventory drops to zero just before an order arrives– Variables:

• S = order cost ( per order)• H = carrying cost (per item per order)• D = annual demand• Order cost = (D/Q)/S; Carrying Cost = (Q/2)*H

Q* = sqrt(2DS/H)

• Q* = quantity to order that will minimize cost under the EOQ model

EOQ = Economic Order Quantity

• Simulation modeling is a flexible modeling approach that is capable of replicating the real world intricacies of an inventory system but it is also generally an expensive (time and money) approach. In the previous worksheet we used a historical simulation to find a good policy (values for Q and ROP). We found the policy by trial and error, facilitated by a two-way data table.

• We will use a simpler approximate two-step analytical method to first find Q and then find ROP. We use the well-known EOQ model (which trades off ordering costs with holding costs) to find Q. Then we use this Q and an estimate of the probability distribution of demand during the lead time to determine ROP, either by meeting a pre-specified level of service or fill rate or by minimizing the costs of incurring a shortage plus the cost of carrying extra safety stock). This two-step method involves many approximations, but in practice it usually gives a near-optimal policy

• Find Q*, then use LTD (lead time demand model) to find ROP using Q from previous step.

Simulation versus EOQ

Dimension Simulation EOQ + LTD

Ease of evaluating a policy

Need to build model – time consuming

Simple formula for RC – back of an envelope

Finding the optimum Trial and error / data table

Plug into formula for Q*

Random demand fluctuations

Taken into account Ignored in EOQ

Seasonal demand fluctuations

Can be taken into account

Ignored

Shortages Taken into account Ignored in EOQ

Likely errors

(common mistakes)

Errors in formulas Inconsistent units

pg. 151

LTD – Lead time demand

• Lead time = how long we wait while receiving an order

• Lead time demand = how much demand would occur while we are waiting for our order to arrive

• Goal is to reduce the probability of shortages

LTD – Lead time demand

• Set the lead time• LTD will be the sum of the demand for the lead

time• Shortage will occur if LTD>=ROP

– If(LTD>=ROP,LTD-ROP,0)• For shortage per cycle

– cycle= demand/Q– Shortage = cycle*average shortages/year– Fill rate

• = sales/demand • = (demand-shortages)/demand