Project and Production Management
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Project and Production Management
Module 8
Production Planning over the Short Term Horizon
Prof Arun Kanda & Prof S.G. Deshmukh, Department of Mechanical Engineering,Indian Institute of Technology, Delhi
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MODULE 8: Production Planning over the Short
Term Horizon1. Forecasting
2. The Analysis of Time Series
3. Aggregate Production Planning: Basic Concepts
4. Aggregate Production Planning: Modelling approaches
5. Illustrative Examples
6. Self Evaluation Quiz
7. Problems for Practice
8. Further exploration
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1.Forecasting
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FORECASTING
Forecasting is essential for a number of planning decisions
LONG TERM DECISIONS New Product Introduction Plant Expansion
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MEDIUM TERM DECISIONS Aggregate Production Planning Manpower Planning Inventory Policy
SHORT TERM DECISIONS Production planning Scheduling of job orders
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PLANNING PROCESSA Forecast of Demand is an essential Input for
PlanningSystem
Objectives
System to beManaged
Constraints- Budget / Space
Resources - Men
- Equipment
DemandForecast
Plan ofAction
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METHODS OF FORECASTING
(a) Subjective or intuitive methods Opinion polls, interviews DELPHI
(b) Methods based on averaging of past data Moving averages Exponential Smoothing
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(c) Regression models on historical data Trend extrapolation
(d) Causal or econometric models
(e) Time - series analysis using stochastic models
Box Jenkins model
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FORECASTING Objective Scientific Free from ‘BIAS’ Reproducible Error Analysis Possible
PREDICTION Subjective Intuitive Individual BIAS Non - Reproducible Error Analysis Limited
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COMMONLY OBSERVED “NORMAL” DEMAND
PATTERNS
D
t
Constant
D
t
LinearTrend
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D
t
CyclicD
t
SeasonalPattern with
Growth
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ABNORMAL DEMAND PATTERNS
TransientImpulse
SuddenRise
SuddenFall
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OPINION POLLSPersonal interviews
e.g. aggregation of opinion of sales representatives to obtain sales forecast of a region
Knowledge base (experience) Subjective bias
Questionnaire method questionnaire design choice of respondents obtaining respondents analysis and presentation of results (forecasting)
Telephonic conversation Fast
DELPHI
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DELPHIA structured method of obtaining responses from
experts.
Utilizes the vast knowledge base of experts
Eliminates subjective bias and ‘influencing’ by members through anonymity
Iterative in character with statistical summary at end of each round (Generally 3 rounds)
Consensus (or Divergent Viewpoints)
usually emerge at the end of the exercise.
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Coordinator
Expert 1
Expert 2
Expert n
1990 1995 2000 2005 20101990
year
• Mean•Median•Std. deviation
A Statisticalsummary
can be givenat end of
each round
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DELPHI (Contd.)
Round1
Round2
Round3
MovingTowards
Consensus
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DELPHI (Contd.)
Round1
Round2
Round3
MovingTowardsDivergent
View Points
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MOVING AVERAGESMonth Demand 3 Month MA 6 Month MA
Jan 199 Feb 202 Mar 199 200.00 Apr 208 203.00 May 212 206.33 Jun 194 203.66 202.33 Jul 214 205.66 207.83 Aug 220 208.33 210.83 Sep 219 216.66 213.13
Oct 234 223.33 217.46 Nov 219 223.00 218.63 Dec 233 227.66 225.13
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K PERIOD MA = AVERAGE OF K MOST RECENT OBSERVATIONS
For instance : 3 PERIOD MA FOR MAY
= Demands of Mar, Apr, May / 3= (199 + 208 + 121) / 3 = 206.33
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CHARACTERISTICS OF MOVING AVERAGES
Dt
t
Dt
t
(1) MOVING AVERAGES LAG A TREND
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Dt
t
(2) MOVING AVERAGES ARE OUT OF PHASE
FOR CYCLIC DEMAND
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Dt
t
(3) MOVING AVERAGES FLATTEN PEAKS
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EXPONENTIAL SMOOTHINGFt = one period ahead forecast made at
time time t
Dt = actual demand for period t
= Smoothing constant (between 0 & 1)
(generally chosen values tie between 0.01 and 0.3)
Ft = Ft-1 + (Dt - Ft-1)
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Ft = Dt +(1 - ) Ft-1
= Dt +(1 - ) [ Dt-1 +(1 - )2 Ft-2 ]
=……..
= [Dt +(1 - ) Dt-1 +(1 - )2 Dt-2 + …..
+ (1 - )t-1 D1 + (1 - )t F0]
(1- ) (1- )2
tt-1t-2
Weightages given to past data decline exponentially.
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MOVING AVERAGESAND
EXPONENTIAL SMOOTHING
(Equivalence between & N :) = 2 / (N+1)
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J F M A M J J A S O N D J190
200
210
220
230
Dem
and
Month
3 Month MA
6 Month MA
= 0.3
= 0.1
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Common Regression Functions
dt
t
dt’ forecast
dt actual demand (for time period t)
dt’ = a + bt (parameters a, b)
Linear
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dt
t
dt’ = a + u Cos (2/n)t + v Sin (2/n)t (parameters a, u, v)
Cyclic
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dt
t
dt’ = a +bt + u Cos (2/n)t + v Sin (2/n)t (parameters a,b,u, v)
Cyclic withGrowth
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dt
tdt’ = a +bt + ct2 (parameters a,b,c)
Parameters Determined by Minimizing the Sum of Squares of errors,
Quadratic
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REGRESSION
J F M A M J J A S O N D J190
200
210
220
230
Dem
and
Month (t)
Actual Data 218
246
Ft = 193 + 3t(Regression Line)
Forecast for next
JAN
1 2 3 4 5 6 7 8 9 10 11 12 13
232
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Standard error of estimate = (Dt - Ft)2
Where
Dt = actual demand for period t = 7.32
Ft = forecast for period t
n = no. of data points
f = degrees of freedom lost (2 in this case)
95 % confidence limits for forecast of next JAN ~ 232 14 (* 2 sigma)
n
t = 1 n - f
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CAUSAL MODELSHere demand is related to
Causal variables GNP Per Capita income Consumer Price index …………
Demand for tyres
= f (Production of new automobiles, Replacements by existing autos,
Govt policy on automobiles, …..)
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Dt = Pt + Pt-5 +
could be a simplified causal model
(Here parameters , ,, are estimated by regression from data)
For a Causal Model to be Useful
The causal variables should be
Leading
Highly correlated with the variable of interest
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TIME SERIES ANALYSISTime series decomposed into
Trend Seasonality Cycle Randomness
And Forecast generated from these components
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Stochastic modelling ( Box and Jenkins)
Various processes eg.Autoregressive (AR) order pMoving average (MA) order qARMA order (p,q)ARIMA order (p,d,q)
are used to fit the most appropriate model.These models are accurate (for short term demand forecasting) but highly cumbersome to develop.
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Past Data
ForecastGeneration
ManagerialJudgement
&Experience
ForecastControl
Current Data
ModifiedForecast
Forecasting System
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Moving Range Chart to Control Forecasts
MR = | (Ft -Dt) - (Ft-1 - Dt-1) |
(Moving Range)
MR = MR / (n – 1) ( There are n-1 moving ranges for n period)
Upper Control Limit (UCL) = + 2.66 MR
Lower Control Limit ( LML) = - 2.66 MR
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VARIABLE TO BE PLOTTED = (Ft - Dt)
01020
30
-10
20
-20
30
-30 Month
(Control Chart for Example)
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SUMMARY
Importance of forecasting in planning
Various Methods of forecasting Subjective methods like opinion polls & Delphi Moving Averages & Exponential Smoothing Trend extrapolation by regression Causal models Time series decomposition
Forecast Control
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2. The Analysis of Time Series
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CORRELATION
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CORRELATION vs REGRESSION?
Correlation examines if there is an association between two variables,
and if so to what extent.
Regression establishes an appropriate relationship between the variables
X
Y
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r = 0
SCATTER DIAGRAM
*
*
*
** *
** *
***
**
**
* *
**
*
*
** * *
r > 0 r < 0
Positive correlation Negative correlation
No correlation Non-linear association
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THE CORRELATION COEFFICIENT
Pearson’s correlation coefficient, r
= (1/n) Sum [(X- X) (Y-Y)]
sigma X sigma Y
(The numerator is the
Co-variance between X and Y)
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METHODS OF COMPUTATION
Direct computations using the formulaCumbersome and lengthy computations
Short-cut or the U-V method Involves any conveniently assumed meanSuitable scaling of variables
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S. No. X Y x=X-X
y=Y-Y
x2 y2 xy
1 50 700 21 274 441 75,076 5,754
2 50 650 21 274 441 50,176 4,704
3 50 600 21 174 441 30,276 3,654
4 40 500 11 74 121 5,476 814
5 30 450 1 24 1 576 246 20 400 -9 -26 81 676 2347 20 300 -9 -126 81 15,876 1,134
8 15 250 -14 -176 196 30,976 2,464
9 10 210 -19 -216 361 46,656 4,104
10 5 200 -24 -226 576 51,076 5,424
Total 290 4260 0 0 2,740 3,06,840
28,310
Advertisement expenditure (X) vs Sales (Y) figures for 10 years in Lacs of Rupees.
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X = 290/10 = 29 : Y = 4260/10 =426 r = Σxy /[ Σx2Σy2]1/2 = 28310/ (2740 * 306840)1/2 = 0.976 Coefficient of Determination = r2 = 0.953
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WHAT IS REGRESSION?
Discovering how a dependent variable (Y) is related to one or more independent variables (X)
Y
X
Y = f(X)
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CRITERION FOR BEST FIT?
Y = f(X)
Mean error Minimize ? Mean absolute error Sum of Squares of Errors
Least Squares Criterion is the generally preferred criterion
Positiveerror
Negativeerror
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FIITING IN A STRAIGHT LINE
• Ft = a + bt is the equation of the line to be fitted• Ft is the fitted function for time t• Dt is the actual demand for period t• Past data is available for n periods• Parameters a & b have to be estimated from the data using least squares criterion
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LEAST SQUARES NORMAL EQUATIONS
SSE = Σ(Dt – Ft)2 = Σ (Dt- a – bt)2
To minimize (SSE)
d(SSE)/da = Σ 2(Dt – a –bt)(-1) = 0
d(SSE)/db = Σ 2(Dt – a – bt)(-t) = 0
Or a (n) + b (Σ t) = Σ Dt
a(Σt) + b (Σ t2) = Σ t Dt
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LEAST SQUARES NORMAL EQUATIONS (2)
These are two linear simultaneous equations in the two unknown parameters a and b which can be solved by any of the well known methods eg Cramer’s Rule.
These equations are called
Least Squares Normal Equations
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a (n) + b (Σ t) = Σ Dt ( Least Squares Normal
a(Σt) + b (Σ t2) = Σ t Dt Equations)
Σ Dt Σ t
a = Σ tDt Σt2 = ΣDt Σt2 – Σt ΣtDt
n Σt n Σt2 – (Σt)2
Σt Σt2
n ΣDt
b = Σt ΣtDt = n ΣtDt - Σt ΣDt
n Σt n Σt2 – (Σt)2
Σt Σt2
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ORGANIZING COMPUTATIONS
S. No ti Di tiDi ti2
1 t1 D1 t1D1 t12
2 t2 D2 t2D2 t22
n tn Dn tndn tn2
Totals ∑ti ∑ Di ∑tiDi ∑ti2
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COMPUTATIONAL SIMPLIFICATIONS
By choosing an origin and scale of data such that
Σ t = 0
the values of the parameters become
a = Σ Dt / n
b = Σ tDt / Σ t2
(This is useful for equally spaced data with even or odd number of data points)
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DEMAND HISTORYMonth Demand
Jan 199 Feb 202 Mar 199 Apr 208 May 212 Jun 194 Jul 214 Aug 220 Sep 219
Oct 234 Nov 219 Dec 233
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REGRESSION
J F M A M J J A S O N D J190
200
210
220
230
Dem
and
Month (t)
Actual Data 218
246
Ft = 193 + 3t(Regression Line)
Forecast for next
JAN
1 2 3 4 5 6 7 8 9 10 11 12 13
232
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Standard error of estimate = (Dt - Ft)2
= 7.32WhereDt = actual demand for period tFt = forecast for period tn = no. of data pointsf = degrees of freedom lost (2 in this
case)95 % confidence limits for forecast of next
JAN ~ 232 14 (2 sigma limits)
t=1,n
n-f
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Common Regression Functions
dt
t
dt’ forecast
dt actual demand (for time period t)
dt’ = a + bt (parameters a, b)
Linear
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dt
t
dt’ = a + u Cos (2/n)t + v Sin (2/n)t (parameters a, u, v)
Cyclic
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dt
t
dt’ = a +bt + u Cos (2/n)t + v Sin (2/n)t (parameters a,b,u, v)
Cyclic withGrowth
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dt
tdt’ = a +bt + ct2 (parameters a,b,c)
Parameters Determined by Minimizing the Sum of Squares of errors,
Quadratic
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2. Analysis of Time series
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COMPONENTS OF A TIME SERIES
Trend, Tt
Seasonality St
Cycle Ct
Randomness Rt
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MULTIPLICATIVE MODEL
Xt = Tt * St * Ct * Rt
Time t Identify
Tt
St
Ct
Rt
Obtain Xt
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DE-SEASONALIZING THE TIME SERIES
If the time series represents a seasonal pattern of L periods, then by taking a moving average Mt of L periods, we would get the mean value for the year. This would be free of seasonality and contain little randomness (owing to averaging)
Thus Mt = Tt * Ct
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DETERMINATIN OF THE TREND
To the de-seasonalized series, a suitable trend line could be fitted using RegressionThe choices could beLinearQuadraticExponentialOther
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ESTIMATING THE CYCLE COMPONENT
After the Trend Tt has been estimated
one can use
Ct = Mt / Tt
to estimate the Cycle Component, Ct
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DETERMINATION OF SEASONAL INDICES
To isolate seasonality, one could simply divide the original series by the moving average
Xt/ Mt = Tt *St* Ct* Rt /Tt *Ct
= St * Rt
Averaging over same months eliminates randomness and yields seasonality indices
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PROCEDURE FOR DECOMPOSTION
1 Decompose the time series into its components Find seasonal component Deseasonalize the demand Find trend component
2 Forecast future values of each component Project trend component into the future Multiply trend component by seasonal component
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EXAMPLE 1
Past Sales Average Sales Seasonal Factor (1000/4)
Spring200 250 200/250 = 0.8Summer 350 250 350/250 = 1.4Fall 300 250 300/250 = 1.2Winter150 250 150/250 = 0.6
Total 1000
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EXAMPLE 1(contd)
Expected Average Sales Seasonal Factor next year demand (1100/4) Next year forecastSpring 275 * 0.8 = 220Summer 275 * 1.4 = 385Fall 275 * 1.2 = 330Winter 275 * 0.6 = 165
Total 1100
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EXAMPLE 2
Quarter Amount
I- 2000 300II- 2000 200III- 2000 220IV- 2000 530
Quarter Amount
I - 2001 520II- 2001 420III- 2001 400IV- 2001 700
Computing Trend & Seasonal Factor
on a 2 year demand history
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EXAMPLE 2 (contd 1)
Quarter Demand
2000I 300II 200III 220IV 5302001I 520II 420III 400IV 700
From Trend Equation Ratio of Seasonal Tt = 170+55t Actual / Factor Trend
225 1.33 I 1.25 280 0.71 II 0.78335 0.66 III 0.69390 1.36 IV 1.25
445 1.17500 0.84555 0.72610 1.15
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EXAMPLE 2 (contd 2)Forecast for 2002 using Trend and Seasonal factors
I – 2002 [170+ 55*09] 1.25 = 831II –2002 [170+ 55*10] 0.78 = 562III-2002 [170+ 55*11] 0.69 = 535IV- 2002 [170 + 55*12] 1.25 = 1,038
Trend * Seasonal factor = Forecast
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EXAMPLE 3
For the given demand history prepare a forecast using decomposition
Period Actual Period Actual
1 300 5 416 2 540 6 7603 885 7 11914 580 8 760
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EXAMPLE 3 (Contd 1)Period Actual Period Seasonal Deseasonalized x Y Average Factor Demand
1 300 358 0.527 568.992 540 650 0.957 564.093 885 1038 1.529 578.924 580 670 0.987 587.795 416 0.527 789.016 760 0.957 793.917 1191 1.529 779.088 760 0.987 770.21Total 5432 2716 8.0Average 679 679 1.0
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EXAMPLE 3 (Contd 2)
Period x Deseazonalized demand, y x2 xy 1 568.99 1 569.0 2 564.09 4 1128.23 578.92 9 1736.74 587.79 16 2351.25 789.01 25 3945.0 6 793.91 36 4763.47 779.08 49 5453.68 770.21 64 6161.7Su ms 5432 204 26,108.8Average 679
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EXAMPLE 3 (Contd 3)The regression equation for deseasonalized data:
26108 – (8) (4.5) (679) = 39.64 (slope of st. line) 9204) – (8) )(4.5)2
a = Y – bx = 679- 39.64(4.5) = 500.6 (intercept of st.line)
Thus, Y = 500.6+ 39.64x
is the result of the deseasonalized regression line
b =
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EXAMPLE 3 (Contd 4)
Forecasts for the next four quarters of the following year
Period Trend Seasonal Final
Forecast Factor Forecast
9 857.4 * 0.527 = 452.0
10 897.0 * 0.957 = 858.7
11 936.7 * 1.529 = 1431.9
12 976.3 * 0.987 = 963.4
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APPLICATION OF AUTO-CORRELATION
Random data
Trend
Seasonal
lag
lag
lag
Autocorrelation (range –1 to 1) plotted on the x-axis
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CONCLUSIONS
Choice of forecasting technique Problem context Accuracy desired Cost Planning horizon
Importance of Correlation and Regression in analysis of Time Series Least Squares Normal Equations with examples
Time series decomposition Components (Trend, seasonal, cycle & random) Deseasonalization Re-construction of the time series
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CONCLUSIONS (Contd)
Three illustrative examples of decomposition
Use of Auto-correlations to identify the kind of time seriesRandomTrend Seasonality
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3. Aggregate Production Planning: Basic Concepts
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AGGREGATE PRODUCTION PLANNING
Concerned with planning overall production of
all products combined (in tonnes of steel,
litres of paint etc.) Over a planning horizon
(generally next 3 to 6 months) for a given
(forecast) demand schedule.
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CapacitiesCost
Commitments
Forecast ofDemand
Aggregate Production
Plan
Work force
M/c TimeAllocation over planning horizon
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MANAGEMENT OPTIONS TO MEET FLUCTUATING
DEMANDBuild inventories in slack periods in anticipation of higher demands later in planning horizon.Carry backorders or tolerate lost sales during peak periods.Use over time in peak periods, under time in slack periods to vary output, while holding work force and facilities constant
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Vary capacity by changing size of work force through hiring and firing
Vary capacity through changes in plant and equipment (generally long term option)
Each option involves cost (tangible or intangible). Aim in aggregate production planning is to choose best option.
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KINDS OF COSTS INVOLVED
Procurement Costs
Production Costs
Inventory holding Costs
Shortage losses associated with backorders and lost sales
Costs of increasing / decreasing work force
Cost of overtime / under time
Cost of changing production rates (Set ups, opportunity losses etc)
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Period Expected Demand
Cumulative Demand
1 100 100
2 180 280
3 220 500
4 150 650
5 100 950
6 200 950
7 250 1200
8 300 1500
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Period Expected Demand
Cumulative Demand
9 260 1760
10 250 2010
11 240 2250
12 210 2460
13 140 2600
EXPECTED SALES FOR A ONE YEAR PLANNING HORIZON BROKEN INTO 13 (4 WEEK) PERIODS
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GRAPHICAL PROCEDURE
1 2 3 4 5 6 7 8 9 10 11 12 13
500
1000
1500
2000
2500
3000
0
Plan 1 - - Constant Production: 200/period
Plan 2 - - Varying Production. 150 /period Period 1 - 5
250 /period Period 6 - 11
175 /period Period 12 - 13
Actual Cumulative Demand
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Analysis of plan1Period Prod. Inv. Back
Order Capacity Change
Over Time
Sub Contract
1 200 100 0 +20 0 0
2 200 120 0 0 0 0
3 200 100 0 0 0 0
4 200 150 0 0 0 0
5 200 250 0 0 0 0
6 200 250 0 0 0 0
7 200 200 0 0 0 0
8 200 100 0 0 0 0
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Analysis of plan1
Period Prod. Inv. Back Order
Capacity Change
Over Time
Sub Contract
9 200 40 0 0 0 0
10 200 0 10 0 0 0
11 200 0 50 0 0 0
12 200 0 60 0 0 0
13 200 0 0 0 0 0
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Analysis of plan2Period Prod. Inv. Back
Order Capacity Change
Over Time
Sub Contract
1 150 50 0 -30 0 0
2 150 20 0 0 0 0
3 150 0 50 0 0 0
4 150 0 50 0 0 0
5 150 0 0 0 0 0
6 250 50 0 +50 40 10
7 250 50 0 0 40 10
8 250 100 0 0 40 10
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Analysis of plan2
Period Prod. Inv. Back Order
Capacity Change
Over Time
Sub Contract
9 250 0 10 0 40 10
10 250 0 10 0 40 10
11 250 0 0 0 40 10
12 175 0 35 -25 0 0
13 175 0 0 0 0 0
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ASSUMPTIONSAll shortages backlogged
Regular Time Capacity = 200 units/period
Max. Overtime = 20% of Regular Time Capacity
Overtime Preferable to Subcontract
Assumed Initial Inventory = 0
Initial Regular Time Prodn. Capacity = 180
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NATURE OF COSTS AND SOLUTION PROCEDURES
Production Inventory Cost Linear Cost
Cost
•LP
•Simplified Transportation
•Transportation
Xt It
Ct (
Xt)
0
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NATURE OF COSTS AND SOLUTION PROCEDURES
Production Inventory Cost Convex Cost
Cost
•Transportation
(after piece wise linearization
•HMMS ( LDR)
(quadratic cost)Xt It
Ct (
Xt)
0
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NATURE OF COSTS AND SOLUTION PROCEDURES
Production Inventory CostConcave and Cost
Piecewise Concave Cost
•Dynamic Programming
(Shortest path Wangner / Whitin)Xt It
Ct (
Xt)
0
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LINEAR DECISION RULESHolt, Modigliani, Muth & Simon (HMMS) 1955
(Study in large paint manufacturing unit)
Wt = Work force level in period t, t =1,…..T
Xt = Aggregate prodn.level in period t
It = Actual aggregate net inventory at end of period t
It* = Desired (ideal) aggregate net inventory at end of period t
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COSTS INCURRED IN PERIOD IN PERIOD t
1. Regular time payroll cost
C1Wt + C13
2. Work force change cost
C2 (Wt - Wt-1 - C11 )2
Wt
Wt - Wt-1
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3. Overtime Cost
C3 (Xt - C4 Wt)2 + C5Xt - C6 X Wt + C12XtWt
4. Inventory related costs
C7 (It - It*)2 = C7 (It - C8 - C9Dt)2
LDR W1*
X1 *
Linear fns. of Dt, W0, I0
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Other Methods To Handle Aggregate Planning
Linear Programming Formulation(Hanssmann & Hess)
Search Decision Rules(Taubert)
Goal Programming Formulation(Multiple goals)
Parametric production planning(Jones)
Management coefficients model(Bowman)
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SUMMARY
Aggregate Production Planning relevant for fluctuation demands for medium term horizons (6 months -1 year)A simple graphical procedure that generates good solutions by examining the demand pattern was presentedVarious ways to meet a fluctuating demand were consideredA summary of solution procedures including the Linear Decision Rules were indicated.
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4. Aggregate Production Planning:Modelling Approaches
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SOLUTION TECHNIQUES
Linear costsLinear ProgrammingTransportation Model
Piecewise linear and Convex costsHolt, Modigliani, Muth and Simon’s LDRsTransportation Model
Concave and Arbitrary CostsNetwork based ModelNon linear programming
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LP : DEFINITION OF VARIABLES
r,v = cost /unit produced during regular time and overtime respectivelyPt, Ot = units produced during regular time and overtime, respectivelyH,f = hiring and layoff costs per unit, respectivelyAt, Rt = number of units increased or decreased, respectively, during consecutive periodsC = inventory costs [per unit per periodDt = sales forecastMt, Yt = Available regular time and overtime capacities respectively
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LP: OBJECTIVE FUNCTION
Min C (production, hiring, layoffs, overtime, undertime and inventory)
= r Pt + h At + f Rt + v Ot + c It
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LP: CONSTRAINTS
Pt <= Mt, t=1,2, , k
Ot<= Yt, t= 1,2, , k
It=It-1 + Pt +Ot-Dt , t= 1, 2, , k
At >= Pt-Pt-1 t= 1,2, , k
Rt >= Pt-1 – Pt t= 1,2, k
(All variables non-negative)
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TRANSPRTATION MODELPeriod (month) 1 2 3 4Demands 100 105 200 95Production capacity(units)
Regular time 100 80 120 60Overtime 40 40 50 30
Production Costs (Rs) Regular time 16 20 22 18Overtime 24 30 30 26
Holding cost/unit/period (Rs) 2 2 4 5Initial on hand inventory (units) 50 unitsFinal desired inventory (units) 20 units
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D1 D2 D3 D4 Ifinal Dummy I in 50
R1 100
O1 40
R2 80
O2 40
R3 120
O3 50
R4 60
O4 30
100 105 200 95 20 50 570
Setting up the Transportation Problem
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D1 D2 D3 D4 Ifinal Dummy I in 0 2 4 8 13 0
50
R1 16 18 20 24 29 0 100
O1 24 26 28 32 37 0 40
R2 20 22 26 31 0 80
O2 30 32 36 41 0 40
R3 22 26 31 0 120
O3 30 34 39 0 50
R4 18 23 0 60
O4 26 31 0 30
100 105 200 95 20 50 570
Introducing Unit Costs
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D1 D2 D3 D4 Ifinal Dummy I in 0
50 2 4 8 13 0
50/0
R1 16 50
18 20 24 29 0 100/50
O1 24 26 28 32 37 0 40
R2 20 22 26 31 0 80
O2 30 32 36 41 0 40
R3 22 26 31 0 120
O3 30 34 39 0 50
R4 18 23 0 60
O4 26 31 0 30
100 105 200 95 20 50 570
Satisfying 1st Period Demand
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D1 D2 D3 D4 Ifinal Dummy I in 0
50 2 4 8 13 0
50/0
R1 16 50
18 50
20 24 29 0 100/50/0
O1 24 26 28 32 37 0 40
R2 20 55
22 26 31 0 80/25
O2 30 32 36 41 0 40
R3 22 26 31 0 120
O3 30 34 39 0 50
R4 18 23 0 60
O4 26 31 0 30
100 105 200 95 20 50 570
Satisfying 2nd Period Demand
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D1 D2 D3 D4 Ifinal Dummy I in 0
50 2
4 8 13 0
50/0
R1 16 50
18 50
20 24 29 0 100/50/0
O1 24 26
28 40
32 37 0 40/0
R2 20 55
22 25
26 31 0 80/25/0
O2 30
32 36 41 0 40
R3 22 120
26 31 0 120/0
O3 30 15
34 39 0 50/35
R4 18 23 0 60
O4 26 31 0 30
100 105 200 95 20 50 570
Satisfying 3rd Period Demand
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D1 D2 D3 D4 Ifinal Dummy I in 0
50 2 4 8 13 0
50/0
R1 16 50
18 50
20 24 29 0 100/50/0
O1 24 26 28 40
32 37 0 40/0
R2 20 55
22 25
26 31 0 80/25/0
O2 30 32 36 41 0 40
R3 22 120
26 31 0 120/0
O3 30 15
34 5
39 0 50/35/30
R4 18 60
23 0 60/0
O4 26 30
31 0 30/0
100 105 200 95 20 50 570
Satisfying 4th Period Demand module 8: Production Planning
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D1 D2 D3 D4 Ifinal Dummy I in 0
50 2 4 8 13 0
50/0
R1 16 50
18 50
20 24 29 0 100/50/0
O1 24 26 28 40
32 37 0 40/0
R2 20 55
22 25
26 31 0 80/25/0
O2 30 32 36 41 0 40
R3 22 120
26 31 0 120/0
O3 30 15
34 5
39 20
0 50/35/30/10
R4 18 60
23 0 60/0
O4 26 30
31 0 30/0
100 105 200 95 20 50 570
Satisfying final inventory restrictions module 8: Production Planning
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D1 D2 D3 D4 Ifinal Dummy I in 0
50 2 4 8 13 0
50/0
R1 16 50
18 50
20 24 29 0 100/50/0
O1 24 26 28 40
32 37 0 40/0
R2 20 55
22 25
26 31 0 80/25/0
O2 30 32 36 41 0 40
40/0
R3 22 120
26 31 0 120/0
O3 30 15
34 5
39 20
0 10
50/35/30/10/0
R4 18 60
23 0 60/0
O4 26 30
31 0 30/0
100 105 200 95 20 50 570
Satisfying Dummy restrictions
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OPTIMAL SOLUTION
Total Cost of optimal solution = Sum of (unit cost x
quantity) = Rs 10370 for the planning horizon
This includes the costs of production on regular and overtime and the costs of holding inventories
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OPTIMAL PRODUCTION PLAN
Period 1
Period 2 Period 3 Period 4
Regular time production
100 80 120 60
Overtime production
40 ----
(-40)
40
(-10)
30
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NETWORK FLOW PROCEDURE
General cost structure
Linear Non-linear arbitrary Concave with set ups
Nature of Inventory Holding and Shortage Costs
It ItIt
cost
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PRODUCTION COST
Qty Qty Qty
Cos
t of
Pro
duct
ion
Linear Piecewise linear Piecewise linear convex concave with set up
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EXAMPLE PROBLEM
Four period problem (t = 1, 2, 3, 4)Production in period t involves a set up cost At
and a unit variable cost ct for raw materials, utilities, labour etc
Ct(Xt) = At + ct Xt , if Xt >0 = 0, if Xt =0
Linear holding costs (no shortage)
At
ct
Xt
Ct(
Xt)
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PROBLEM DATA
Period 1 Period 2 Period 3 Period 4
Demand
(units)10 20 5 15
Set up
(Rs)100 120 120 140
Variable cost (Rs/unit)
8 9 10 10
Holding cost (Rs/unit)
2 4 5 7
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SETTING UP THE NETWORK
0 1 2 3 4
For N periods there will be N+1 nodes and N(N+1)/2 arcs, withN emanating from Node 0, N-1 from Node 1,…,1 from Node N-1
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ARC INTERPRETATIONS
Mjk = Cost of producing in period (j+1) for
the requirements of the periods
(j+1), (j+2), … (k). (j<k)
j k
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COST COMPUTATIONS I
M01= A1+c1(D1)
M02 =A1 + c1 (D1+D2) +h1(D2)
M03 =A1 + c1 (D1+D2+D3) + h1 (D2+D3)+ h2 (D3)
M04 = A1 +c1 (D1+D2+ D3+D4) + h1 (D2+D3+ D4) +
h2(D3+ D4) + h3( D4)
M12 = A2 +c2(D2)
M13 = A2 +c2(D2+ D3) + h2(D3)
M14= A2 + c2 (D2+D3+D4)+ h2 (D3+D4) +h3 (D4)
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COST COMPUTATIONS II
M23 = A3 + c3(D3)
M24 = A3 + c3(D3+D4) + h3 (D4)
M34 = A4 + c4 (D4)
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COMPUTATION OF ARC LENGTHS
M01 = 100+ 8(10) = 180
M02 = 100+8(30) +2(20) = 380
M03 = 100+8(35) +2(25) + 4(5) = 450
M04 = 100 + 8(50) +2(40) +4(20) +5(15) = 735
M12= 120 +9(20)= 300
M13 = 120+9(20)+ 4(5) = 365
M14 = 120 + 9(40) +4(20) +5(15) = 635
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COMPUTATION OF ARC LENGTHS
M23 = 120+ 10(5) = 170
M24 = 120+10(20) +5(15) = 395
M34 = 140+10(15) = 290
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SETTING UP THE NETWORK
0 1 2 3 4M01=180
M02= 380
M03= 450
M04= 735
M12= 300
M13=365
M14 = 635
M23=170
M24= 395
M34=290
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DETERMINING THE SHORTEST PATH IN THE
NETWORK
0 1 2 3 4M01=180
M02= 380
M03= 450
M04= 735
M12= 300
M13=365
M14 = 635
M23=170
M24= 395
M34=290
0 180 380 450 735
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DETERMINING THE OPTIMAL PRODUCTION PLAN
The Shortest path is M04 which yields the optimal production plan with a total cost of Rs 735.
Period
1 2 3 4
Production
50 -- -- --
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ARC INTERPRETATIONS(for the problem with shortages)
Mjk = Minimum Cost of producing for the
requirements of the periods
(j+1), (j+2), … (k). (j<k)
(Notice that production may take place
in any period between (j+1), (j+2),… (k) )
j k
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SUMMARY
LP is a versatile tool to handle Aggregate Production Planning Problems with a variety of linear costs and constraintsThe Transportation Model is capable of dealing with problems with piecewise linear convex costs (regular time, overtime, subcontracting) No shortage case handled by a simple greedy procedure Case with shortages treated by regular transportation
problem procedure
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SUMMARY II
The case of concave and arbitrary costs can be solved using a shortest path procedure on a network. Both cases with and without shortages
can be handled
Sample problems for both procedures were solved to illustrate the procedures
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