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Exam 2 Formulas
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
Chapter 12 Inventory Management
ABC Classification rule:
Class A: ~15% of items, 70-80% annual $ usage
Class B: ~30% of items, 15-25% annual $ usage
Class C: ~55% of items, 5% annual $ usage
Item
$ Usage
% of $ usage
Basic EOQ Model
2 DS
Q* 
H
Cumulative % of $
Cumulative % of no. of items
where, D = Demand per year
S = Ordering cost for each order
H = Holding (carrying) cost per unit per year
Expected number of orders (N) = D/Q
Expected time between orders (T) = (Q/D) No. of days per year = Q/d
Annual ordering cost = NS = (D/Q)S
Annual carrying cost = (Q/2)H
Total annual cost (TC) = (D/Q)S + (Q/2)H
Class
POQ Model
2 DS
H (1  d / p )
Q 
*
p
where, D = Demand per year
S = Ordering cost for each order
H = Holding (carrying) cost per unit per year
p = Daily production rate
d = Daily demand rate = D/No. of working days
Length of production run (t) = Q/p
Rate of increase of inventory during production = (p - d)
Maximum inventory = Imax = (Q/p)(p-d)
Average inventory = Imax/2
Expected number of batches (N) = D/Q
Expected time between orders (T) = (Q/D) or No. of days per year = Q/d
Annual setup cost = NS = (D/ Q)S
Annual carrying cost = (Imax/2)H
Total annual cost (TC) = (D/Q)S + (Imax /2)H
Quantity discount model
Q
where, D = Demand per year
S = Ordering cost for each order
IP = H = Holding (carrying) cost per unit per year
I = Holding cost as a % of item cost
P = Item cost per unit
2 DS
IP
Step 1: Determine Candidate Q
1. Compute Formula-Q for each price break price.
2. If Formula Q < Lower limit for price, then Candidate Q = Lower limit
If Formula Q is within the limits for the price, then Candidate Q = Formula Q
If Formula Q > Upper limit for price, then no candidate Q, ignore the Formula Q
Q-Range
Price
Holding cost/unit = % x P
Formula Q
Adjusted Q
Step 2: Compute total annual cost (TC) for each valid candidate Q and select the candidate Q with
least cost as EOQ.
Total annual cost = Annual holding cost + Annual ordering cost + Annual item cost
i.e. = (Q/2)H + (D/Q)S + PD, where P = cost of the item per unit
ROP Models
Discrete Probability model
Total cost = Annual Holding cost + Annual stock out cost
Annual Holding cost = Safety stock x H
Annual stock-out cost = Expected stock out per cycle x N x Cs
Where, Expected stock out =  (Stock out x Probability)
N = No. of orders per year = D/Q
Cs = Cost of stock out per unit
Reorder point model with Normal distribution:
Reorder point (ROP) = Average demand during lead time + Safety stock
i.e. ROP = d x L + Z dLT
where, d = Demand rate per period
L = Lead time
Z = Normal table value for the given service level
dLT= Standard deviation of demand during lead time (as give in table below)
Lead time is constant
Lead time is variable
Demand is constant
𝜎dLT = 0
𝜎dLT = d𝜎𝐿
Demand is variable
𝜎dLT = d√𝐿
𝜎dLT = √𝐿𝜎𝑑2 + 𝑑 𝜎𝐿2
2
Single-Period model
𝐶
Service level = 𝐶 +𝑠𝐶 , where Cs = Cost of shortage, Co = cost of overage
𝑠
𝑜
Cs = Lost profit = Selling price per unit – Cost per unit
Co = Cost/unit – salvage value/unit
Order quantity =  + Z, where  = mean demand,  = standard deviation of demand
Stock-out risk= 1 - service level