Operations Research & Production Management – Structured Study Document
1. Operations Research
1.1 PERT & CPM
Distribution in PERT
- Activity duration → Beta Distribution (skewed)
- Project completion time → Normal Distribution
- Reason: Sum of multiple activities follows Central Limit Theorem
Expected Time (PERT)
T_e = \frac{t_o + 4t_m + t_p}{6}
Variance
\sigma^2 = \left(\frac{t_p - t_o}{6}\right)^2
Standard Deviation of Project
\sigma = \sqrt{\sum \sigma^2}
Critical Path
- Longest path in the network
- Determines project duration
Steps:
- Forward pass:
E_j = \max(E_i + t_{ij})
- Backward pass:
L_i = \min(L_j - t_{ij})
- Slack:
- Head slack =
- Tail slack =
- Critical activities: Slack = 0
Shortest Path
S_G = \min(S_O + d_{OG},\; S_R + d_{RG})
Crashing
- Reduce project duration at minimum cost
\text{Cost Slope} = \frac{Crash\ Cost - Normal\ Cost}{Normal\ Time - Crash\ Time}
Steps:
- Identify critical path
- Select activity with minimum cost slope
- Crash step-by-step
- Recalculate path
1.2 Linear Programming
Simplex Method
- Evaluates only corner (extreme) points
Alternate Solution
- Occurs when zero appears in non-basic variable
Dual Formulation
- Max ↔ Min
- Constraints ↔ Variables
1.3 Queuing Theory
Inter-arrival Time Distribution
f(t) = \lambda e^{-\lambda t}
Little’s Law
L = \lambda W
M/M/1 Model
\rho = \frac{\lambda}{\mu}
L_q = \frac{\rho^2}{1-\rho}
W_q = \frac{\lambda}{\mu(\mu - \lambda)}
1.4 Transportation & Assignment
Basic Feasible Solution
m + n - 1
Assignment Method (Hungarian)
- Row reduction
- Column reduction
- Optimal assignment
1.5 Sequencing
SPT Rule
- Shortest Processing Time first
Johnson’s Rule (2 Machines)
- Determine optimal job sequence
2. Production Management
2.1 Inventory Control
EOQ Formula
EOQ = \sqrt{\frac{2DC_o}{C_h}}
JIT System
- Batch size = 1 unit
2.2 Layout Types
- Fixed Position → Large products (e.g., aircraft)
- Product Layout → Assembly line
- Job Shop → High variety, low volume
- Mass Production → High volume, low flexibility
2.3 Forecasting
F_{t+1} = F_t + \alpha(D_t - F_t)
2.4 MRP & Planning
- MRP derived from Master Production Schedule (MPS)
- Chase strategy → Production matches demand
3. Quality Control & Reliability
3.1 Control Charts
- p-chart → Fraction defective
- c-chart → Number of defects
UCL (p-chart)
UCL = \bar{p} + 3\sqrt{\frac{\bar{p}(1-\bar{p})}{n}}
3.2 Reliability
- Series system → Multiply reliabilities
- Parallel system → Complement rule
Maintainability
M(t) = 1 - e^{-t/MTTR}
3.3 Acceptance Sampling
AOQ = \frac{P_a \cdot p (N-n)}{N}
- Producer’s risk → Rejecting good lot
4. Value Engineering
\text{Value} = \frac{\text{Function}}{\text{Cost}}
- Increase value by:
- Increasing function
- Reducing cost
5. Engineering Economy
Straight Line Depreciation
\text{Annual Depreciation} = \frac{Cost - Salvage}{Life}
Sum of Years Digits (SYD)
\text{Depreciation} = \frac{\text{Remaining Life}}{\text{Sum of Years}} \times (Cost - Salvage)
Capital Recovery
CR = P(A/P, i, n)
Capitalized Value
V = \frac{\text{Annual Income}}{i + s}
6. Learning Curve & Work Measurement
- Learning curve reduces time with repetition
- Work measurement → Standard time determination
7. Miscellaneous
- Accident investigation → Fact finding
- EVPI → Maximum value of perfect information
- Normal distribution (±1σ) → 68.3% data
- Replacement decision → Based on minimum average cost
End of Document
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