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MGT 3110 - Exam 1 Formulas
Chapter 1:
Single-factor productivity
Productivi ty 
Units Produced
Input used
Multi-factor productivity
Productivi ty 
Output
Labor  Material  Energy  Capital  Miscellane ous
Change in productivity = New productivity – Base productivity
Productivity growth as a % =
Current productivity−Previous productivity
(
Previous productivity
) x 100
Target productivity = Current productivity x (1 + % improvement)
Target Input = Output / Target productivity
Chapter 3 Project scheduling
Forward pass: ES = Max(EF of all the predecessors); EF = ES + Activity time
Backward pass: LF = Min(LS of all the successors); LS = LF – Activity time
Activity Slack = LS – ES or = LF - EF
PERT
For each activity, te =
𝑎 + 4𝑚 +𝑏
6
𝑏 −𝑎 2
and  = (
2
6
) =
(𝑏 − 𝑎)2
36
TE = Project completion time = = ∑ 𝑡𝑒 of the activities on the critical path
2p = Project variance = ∑ 𝜎 2 of the activities on the critical path;
p = Project standard deviation = √𝑃𝑟𝑜𝑗𝑒𝑐𝑡 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = √𝜎𝑝2
Z=
𝑇 − 𝑇𝐸
𝜎𝑃
, where T = project due date
Project due date (T) for a given probability = TE + z P
CPM
Crash cost per period =
(𝐶𝑟𝑎𝑠ℎ 𝑐𝑜𝑠𝑡 − 𝑁𝑜𝑟𝑚𝑎𝑙 𝑐𝑜𝑠𝑡)
(𝑁𝑜𝑟𝑚𝑎𝑙 𝑡𝑖𝑚𝑒 − 𝐶𝑟𝑎𝑠ℎ 𝑡𝑖𝑚𝑒)
Chapter 4: Forecasting
At = Actual demand for period t
Ft = forecast value for period t
Forecast error Et = At - Ft
Cumulative Forecast Error CFE =  Et
Mean Absolute Deviation (MAD) =
∑ |𝐴𝑡 − 𝐹𝑡 |
𝑛
Mean Absolute Percentage Error (MAPE) =
Mean Square Error = MSE =
Moving Average =
∑(𝐴𝑡 − 𝐹𝑡 )2
∑ |𝐴𝑡 − 𝐹𝑡 |/𝐴𝑡
𝑛
x 100
𝑛
∑ 𝑑𝑒𝑚𝑎𝑛𝑑 𝑖𝑛 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑛 𝑝𝑒𝑟𝑖𝑜𝑑𝑠
𝑛
Weighted moving average =
∑((𝑊𝑒𝑖𝑔ℎ𝑡 𝑓𝑜𝑟 𝑝𝑒𝑟𝑖𝑜𝑑 𝑛)(𝐷𝑒𝑚𝑎𝑛𝑑 𝑖𝑛 𝑝𝑒𝑟𝑖𝑜𝑑 𝑛))
∑ 𝑊𝑒𝑖𝑔ℎ𝑡𝑠
Exponential smoothing: Ft = Ft-1 + (At-1 – Ft-1), 0 <  < 1
Trend projection: ŷ = a + bx
b=
∑ 𝑥𝑦 − 𝑛𝑥̅ 𝑦̅
∑ 𝑥 2 −𝑛𝑥̅ 2
, and a = 𝑦
̅ − 𝑏𝑥̅
Seasonal factor (index) =
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑑𝑒𝑚𝑎𝑛𝑑 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑝𝑒𝑟𝑖𝑜𝑑
𝑂𝑣𝑒𝑟𝑎𝑙𝑙 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑑𝑒𝑚𝑎𝑛𝑑 𝑝𝑒𝑟 𝑝𝑒𝑟𝑖𝑜𝑑
Tracking signal
Et = Error = Demand - Forecast
|Et| = Absolute error = |Demand – Forecast|
CFEt = Running Sum of Et
CAEt = Running Sum of |Et|
MADt = Running MAD for period t = CAEt/ No. of error
Tracking Signal = CFEt/MADt
Associative Model
Y = Variable that needs to be forecasted (Dependent variable or response variable)
X = Variable or factor used to forecast Y (Independent variable or predictor variable)
Forecast = ŷ = a + bx
b=
∑ 𝑥𝑦 − 𝑛𝑥̅ 𝑦̅
∑ 𝑥 2 −𝑛𝑥̅ 2
Sy.x = √
, and a = 𝑦
̅ − 𝑏𝑥̅
∑ 𝑦 2 −𝑎 ∑ 𝑦 − 𝑏 ∑ 𝑥𝑦
𝑛−2
Correlation coefficient (r) =
𝑛 ∑ 𝑥𝑦− ∑ 𝑥 ∑ 𝑦
√[𝑛 ∑ 𝑥 2 −(∑ 𝑥)2 ][𝑛 ∑ 𝑦 2 − (∑ 𝑦)2 ]
Multiple Regression Analysis: ŷ = a + b1x1 + b2x2
Chapter 6: Quality Management
Pareto diagram: Relative frequency=frequency/sum of frequencies
Chapter 6S: Statistical Process Control
̅ -Chart if  is known: 𝑥̿ ± 𝑧σ𝑥̅ , LCL𝑥̅ = 𝑥̿ − 𝑧σ𝑥̅ and UCL𝑥̅ = 𝑥̿ + 𝑧σ𝑥̅
𝑿
where, 𝑥̿ mean of the sample means or a target value set for the process
z = number of standard deviations (2 for 95.45% confidence and 3 for 99.73%)
𝜎𝑥̅ = Standard deviation of sample means =  / n
 = population (process) standard deviation
n = sample size
̅ , LCL𝑥̅ = 𝑥̿ − 𝐴2 𝑅̅ and UCL𝑥̅ = 𝑥̿ ± 𝐴2 𝑅̅
̅ -Chart if  is unknown: 𝑥̿ ± 𝐴2 𝑅
𝑿
∑ 𝑅𝑗
where, 𝑅̅ =
= Average range of samples; Rj = range for one sample
𝑛
A2 = Value found in Table S6.1
𝑥̿ = mean of the sample means
R-Chart
LCLR = D3 , UCLR = D4 ; where, D3 & D4 are values from Table S6.1
p-Chart: 𝑝̅ ± 𝑧𝑝̅
LCLp =
𝑝̅ − 𝑍𝑝̅ , and UCLp =
𝑝̅ + 𝑍𝑝̅
∑ 𝐷𝑒𝑓𝑒𝑐𝑡𝑠
where, 𝑝̅ = mean fraction of defectives in the samples = (𝑁𝑜.𝑜𝑓 𝑠𝑎𝑚𝑝𝑙𝑒𝑠)(𝑛)
z = number of standard deviations (2 for 95.45% confidence and 3 for 99.73%)
𝑝̅ (1−𝑝̅ )
𝜎p = standard deviation of sampling distribution = √
𝑛
p = defectives/n
n = number of observations in each sample
c-Chart: 𝑐̅ ± 𝑧√𝑐̅,
LCLc = 𝑐̅ − 𝑧√𝑐̅ and UCLc = 𝑐̅ + 𝑧√𝑐̅
where, c = number of defectives per unit output
z = number of standard deviations (2 for 95.45% confidence and 3 for 99.73%)
Process Capability
Cp =
𝑈𝑝𝑝𝑒𝑟 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝐿𝑖𝑚𝑖𝑡 − 𝐿𝑜𝑤𝑒𝑟 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝐿𝑖𝑚𝑖𝑡
Cpk = Minimum of {
6𝜎
𝑈𝑝𝑝𝑒𝑟 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑙𝑖𝑚𝑖𝑡 − 𝑥̅ 𝑥̅ − 𝐿𝑜𝑤𝑒𝑟 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑙𝑖𝑚𝑖𝑡
3𝜎
,
 = process standard deviation
Cp and Cpk must be >= 1⅓ for process to be deemed capable,
>=2 for Six-sigma operations
3𝜎
}
Acceptance Sampling
Pa = P(X<= c) = From Poisson table using nPd,
where,
Pa = Probability of accepting the sample
Pd = Probability of defectives in the lot
n = sample size
c = Critical number of defectives in the sample
X = number of defectives in the sample
Producer’s risk = 1 – Pa with Pd = AQL where, AQL = Acceptable Quality Limit
Consumer’s risk = Pa with Pd = LTPD, where, LTPD = Limit Tolerance Percent Defective
Average Outgoing Quality
AOQ =
where,
( Pd )( Pa )( N  n)
N
AOQ = Average Outgoing Quality
Pa = Probability of accepting the sample
Pd = Probability of defectives in the lot
n = sample size
N =Lot size