Reliability Engineering (ME-703 (D)) - Unit 4 Short Notes

1.0 Linear Programming (LP)

1.1 Formulation of LP Problems

• Decision Variables: Quantities to be determined (e.g., units of product A, B).

• Objective Function: Linear function to maximize (profit) or minimize (cost).

  Example: Maximize $$\displaystyle Z = c_1x_1 + c_2x_2 + ... $$

• Constraints: Linear inequalities/equations representing resource limits (man-hours, materials).

• Non-negativity Restrictions: $$\displaystyle x_i \ge 0 $$ for all $i$.

• Standard Form: Maximization problem with ≤ constraints and RHS ≥ 0. Convert ≥ or = using surplus/artificial variables.

[!TIP] Exam Focus: Past papers frequently ask to formulate a real-world problem (product mix, resource allocation) into standard LP form. Identify variables, write objective, list constraints with units.

1.2 Simplex Method

Step-by-Step Algorithm:

1. Convert to standard form (add slack/surplus/artificial variables).

2. Obtain Initial Basic Feasible Solution (IBFS). For ≤ constraints, set non-basic variables = 0, basic variables = RHS.

3. Optimality Test: Check objective row (Cj - Zj) in maximization problem. If all coefficients ≤ 0, current solution is optimal. If any > 0, proceed.

4. Pivot Operation:

   • Entering Variable: Variable with most positive (Cj-Zj).

   • Leaving Variable: Minimum positive ratio (RHS / pivot column coefficient).

   • Perform row operations to update tableau.

5. Repeat until optimality test passed.

Multiple/Optimal Solutions: If a non-basic variable has (Cj-Zj) = 0 at optimum, infinite solutions exist along the edge.

Degeneracy in Simplex:

• Definition: Basic variable becomes zero in a simplex iteration.

• Cause: Tie for minimum ratio during pivot selection.

• Resolution: Use Bland's Rule (choose smallest subscript index for entering/leaving variable) or perturbation method (add tiny ε to RHS).

[!TIP] Common Pitfall: Forgetting to convert ≥ constraints with surplus + artificial variables (requires Big-M or Two-Phase method). Past papers (Dec 2024) test simplex on 2-variable problems—focus on tableau mechanics.

2.0 Transportation Problems

2.1 Problem Structure & Formulation

• Objective: Minimize total transportation cost.

• Balanced: Total row supply = Total column demand.

• Unbalanced: Introduce dummy row (if supply < demand) or dummy column (if demand < supply) with zero cost.

• Decision Variable: $$\displaystyle x_{ij} $$ = units transported from source $i$ to destination $j$.

2.2 Initial Basic Feasible Solution (IBFS) Methods

North-West Corner Rule (NWCR):

1. Start at top-left cell (row 1, col 1).

2. Allocate as much as possible: min(available supply, demand).

3. Adjust supply/demand, move right if demand exhausted, down if supply exhausted.

4. Repeat until all allocations made.

Vogel's Approximation Method (VAM) (High Frequency):

1. For each row/column, compute penalty = difference between two smallest costs.

2. Select row/column with highest penalty.

3. In that row/column, allocate to cell with lowest cost (min(supply, demand)).

4. Adjust supply/demand, cross out exhausted row/col, recalc penalties.

5. Repeat until all allocations made.

[!TIP] VAM gives better IBFS (closer to optimal) than NWCR. Past papers (Jun 2025, Dec 2024, May 2024) consistently test VAM. Show penalty calculations clearly.

2.3 Degeneracy in Transportation

• Definition: Number of positive allocations < $(m + n - 1)$.

• Cause: Simultaneous exhaustion of supply and demand during allocation.

• Resolution: Allocate a very small ε (e.g., 0.001) to an unallocated cell in the same row/column to maintain $(m+n-1)$ basic variables. Do not affect optimality test.

2.4 Transportation with Penalties/Shortages

• Model unfulfilled demand by adding a dummy destination (if penalties on unmet demand) or dummy source (if penalties on unused supply).

• Penalty costs become costs in dummy column/row.

• Solve as standard unbalanced transportation problem.

[!EXAMPLE] Jun 2025: Penalties for unfulfilled demand → add dummy destination with penalty costs as column costs.

3.0 Inventory Management (EOQ Models)

3.1 Fundamental Concepts

• Holding/Carrying Cost (H): Cost per unit per year to hold inventory (storage, insurance, capital).

• Ordering/Setup Cost (S): Fixed cost per order (procurement, transportation).

• Shortage Cost: Cost per unit per year of stock-out (lost sales, backorder).

• Basic EOQ Assumptions: Constant demand rate $D$, instantaneous replenishment, no shortages, fixed ordering cost, constant holding cost.

3.2 Economic Order Quantity (EOQ) Model

Derivation: Minimize Total Annual Cost = Ordering Cost + Holding Cost.

$$TC = \frac{D}{Q}S + \frac{Q}{2}H$$

Differentiate w.r.t $Q$, set = 0:

$$\boxed{EOQ = Q^* = \sqrt{\frac{2DS}{H}}}$$

Key Calculations:

• Optimum Lot Size: $$\displaystyle Q^* $$ from formula.

• Minimum Avg. Yearly Cost: $$\displaystyle TC_{min} = \sqrt{2DSH} $$.

• Optimum Orders/Year: $$\displaystyle N^* = D / Q^* $$.

• Optimum Time Between Orders: $$\displaystyle T^* = 1/N^* $$ (in years) or $$\displaystyle = Q^*/D $$.

3.3 EOQ with Carrying Cost as Percentage

If carrying cost is $i$% of unit cost $C$ per year:
$$\displaystyle H = i \times C $$ (ensure $i$ in decimal, e.g., 8% = 0.08).

3.4 EOQ with Quantity Discounts

Price-Break Model:

1. Calculate EOQ for each price bracket using $$\displaystyle H = i \times C_{bracket} $$.

2. If EOQ for a bracket is within that bracket's quantity range, compute total cost at that EOQ.

3. If EOQ is outside its bracket, compute total cost at the minimum quantity of that bracket.

4. Compare total costs across all feasible brackets. Choose the quantity with lowest total cost.

Total Cost Formula (for given $Q$):

$$TC = \frac{D}{Q}S + \frac{Q}{2}H + D \times C$$

[!TIP] Past papers (Nov 2023) test discount acceptance: Compare total cost at EOQ (without discount) vs. total cost at minimum discount quantity. Find discount % that makes EOQ equal to break quantity by solving $$\displaystyle Q^* = \text{break quantity} $$ for $i$.

4.0 Supply Chain Management (SCM) Core Concepts

4.1 SCM Framework & Flows

• Flows:

  • Material Flow: Physical movement of goods from supplier to customer.

  • Information Flow: Orders, forecasts, inventory levels (bidirectional).

  • Money/Cash Flow: Payments, credit, consignments (reverse direction).

• Objectives: Minimize total system cost, maximize service level.

4.2 Logistics in SCM

• Inbound Logistics: Activities from supplier to company (procurement, receiving, warehousing). Focus: Efficient material inflow.

• Outbound Logistics: Activities from company to customer (distribution, delivery, customer service). Focus: Timely, cost-effective delivery.

[!TIP] Jun 2025 & May 2024 asked importance of outbound logistics → emphasizes customer satisfaction, market reach, competitive advantage.

4.3 The Bull-Whip Effect

• Definition: Demand variability amplifies as one moves upstream (retailer → wholesaler → distributor → manufacturer).

• Causes:

  1. Demand forecasting (using orders, not sales).

  2. Order batching (periodic ordering, quantity discounts).

  3. Price fluctuations (forward buying).

  4. Rationing & gaming (fear of shortages → over-order).

• Implications: Excess inventory, poor customer service, inefficiencies, capacity mismatches.

• Mitigation:

  • Share point-of-sale (POS) data across chain.

  • Vendor-Managed Inventory (VMI).

  • Eliminate incentives causing order amplification.

  • Stabilize prices.

4.4 Cross-Docking

• Definition: Inbound trucks unload directly to outbound trucks with minimal storage (< 24 hrs).

• Importance:

  • Reduces inventory holding costs.

  • Faster throughput, reduced handling.

  • Lower warehousing space needs.

• Disadvantages/Limitations:

  • Requires high coordination & information systems.

  • Needs high, consistent throughput volume.

  • Not suitable for products needing quality checks/long storage.

4.5 Evolution: MRP → ERP → SCM

4.6 Strategic SCM Topics

• Outsourcing: Focus on core competencies, cost reduction, access to expertise.

• Competitive Advantage: Achieved via cost leadership (efficient SCM) or differentiation (superior service).

• Linkage with E-Business: E-procurement, e-commerce integration, real-time data exchange.

• Expenditure & Opportunities: Investment in IT (tracking, forecasting), warehouse automation, strategic partnerships.

5.0 Inventory Classification & Analysis

5.1 ABC Analysis

• Principle: Classify items by annual consumption value = (Annual usage) × (Unit cost). Pareto principle: ~80% value from ~20% items.

• Categories:

  • A-items: High value, low quantity (~15-20% items, ~70-80% value). Tight control, frequent review.

  • B-items: Moderate value/quantity (~30% items, ~15% value). Normal control.

  • C-items: Low value, high quantity (~50-60% items, ~5-10% value). Loose control, bulk ordering.

• Advantages: Focused management, optimized stock levels, reduced administrative cost.

5.2 VED Analysis

• Principle: Classify by Vitality, Essentiality, Desirability (criticality for operations, especially spare parts).

• Categories:

  • V (Vital): Critical for production/safety. High stock, tight control.

  • E (Essential): Important but not critical. Moderate stock.

  • D (Desirable): Nice to have. Low stock, can be delayed.

• Advantages: Prioritizes items for maintenance/production continuity, avoids catastrophic downtime.

5.3 Integration of ABC & VED

• Combined Matrix: 3×3 grid (A/B/C vs V/E/D).

• Critical Spare Parts: A-V items (high value + vital) → highest priority, maximum safety stock.

• Enables nuanced control: e.g., C-V items (low cost but vital) may need higher stock than A-D items (high cost but non-vital).

6.0 Queuing Theory

6.1 Basic Queueing System Components

• Arrival Process: Pattern of customer arrivals (Poisson common).

• Service Mechanism: Number of servers, service time distribution (Exponential common).

• Queue Discipline: Order of service (FCFS most common).

• Capacity: System capacity (finite/infinite).

• Customer Population: Source size (infinite/finite).

6.2 Poisson Arrivals & Exponential Service (M/M/1)

• Notation: M/M/1 → Markovian (Poisson) arrivals, Markovian (Exponential) service, 1 server.

• Parameters:

  • Arrival rate: $\lambda$ (avg. arrivals per unit time).

  • Service rate: $\mu$ (avg. services per unit time, $$\displaystyle \mu > \lambda $$ for stability).

• Exponential Distribution (service time $T$):

$$P(T > t) = e^{-\mu t}$$

Memoryless property: $$\displaystyle P(T > s+t | T > s) = P(T > t) $$.

6.3 Performance Measures & Probability Calculations

• Utilization Factor: $$\displaystyle \rho = \lambda / \mu $$.

• Probability of n customers in system: $$\displaystyle P_n = (1-\rho)\rho^n $$.

• Probability that service time > t: Directly from exponential CDF:

  $$\displaystyle P(T > t) = e^{-\mu t} $$.

• Example (Past papers): "20 customers served per hour" → $$\displaystyle \mu = 20 $$/hr.

  "More than 15 minutes" → $$\displaystyle t = 0.25 $$ hr.

  $$\displaystyle P(T > 0.25) = e^{-20 \times 0.25} = e^{-5} \approx 0.0067 $$.

[!TIP] Convert units consistently: If $\mu$ is per hour, $t$ must be in hours. Past papers (Jun 2025, Dec 2024, May 2024) consistently test this exponential probability calculation.

7.0 Project Management (PERT & CPM)

7.1 Network Diagram Construction

• Activity-on-Arrow (AOA): Arrows represent activities, circles (nodes) represent events (milestones).

• Activity-on-Node (AON): Nodes represent activities, arrows show dependencies (more common now).

• Network Logics (dependencies between activities):

  • FS (Finish-to-Start): Successor starts after predecessor finishes (most common).

  • SS (Start-to-Start): Successor starts after predecessor starts.

  • FF (Finish-to-Finish): Successor finishes after predecessor finishes.

  • SF (Start-to-Finish): Rare.

• Dummy Activity: Zero-duration activity used to show dependency without consuming time/resources (only in AOA).

7.2 Critical Path Method (CPM)

• Deterministic time estimates.

• Forward Pass (Earliest Times):

  • $$\displaystyle EF_i = ES_i + t_i $$

  • $$\displaystyle ES_j = \max(EF_i) $$ for all immediate predecessors $i$ of $j$.

  • Start with $$\displaystyle ES_{start} = 0 $$.

• Backward Pass (Latest Times):

  • $$\displaystyle LS_i = LF_j - t_i $$ for all immediate successors $j$.

  • $$\displaystyle LF_i = \min(LS_j) $$ for all immediate successors $j$.

  • Start with $$\displaystyle LF_{end} = \text{project duration} $$.

• Slack/Float:

  • Total Float = $$\displaystyle LS - ES = LF - EF $$.

• Critical Path: Path with zero total float (longest path). Determines project duration.

7.3 Program Evaluation and Review Technique (PERT)

• Probabilistic time estimates for each activity:

  • Optimistic time ($$\displaystyle t_o $$ or $a$): Minimum time if everything goes well.

  • Pessimistic time ($$\displaystyle t_p $$ or $b$): Maximum time if major problems.

  • Most likely time ($$\displaystyle t_m $$ or $m$): Most realistic estimate.

• Expected Time:

$$\boxed{t_e = \frac{t_o + 4t_m + t_p}{6}}$$

• Variance:

$$\boxed{\sigma^2 = \left(\frac{t_p - t_o}{6}\right)^2}$$

7.4 Project Analysis with PERT

1. Compute $$\displaystyle t_e $$ for all activities.

2. Construct network, perform forward/backward pass on $$\displaystyle t_e $$.

3. Expected Project Duration = $$\displaystyle t_e $$ of terminal event (sum of $$\displaystyle t_e $$ on critical path).

4. Project Variance = Sum of variances ($$\displaystyle \sigma^2 $$) of activities on the critical path.

5. Probability of Completion by Due Date ($D$):

   • Compute Z-score:

$$Z = \frac{D - \text{Expected Duration}}{\sqrt{\text{Project Variance}}}$$

*   Find probability from standard normal table: $P(Z \leq z)$.

[!TIP] Past papers (Nov 2023) ask: "Within how many weeks for 0.99 probability?" → Find $z$ for 0.99 (≈2.33), then $$\displaystyle D = \mu + z\sigma $$. Always use critical path variance only.

7.5 PERT vs. CPM

7.6 Phases of Project Management

1. Conceptualization: Define project, feasibility.

2. Planning: Scope, schedule (PERT/CPM), resources, budget.

3. Execution: Coordinate people/resources, implement plan.

4. Monitoring & Controlling: Track progress, manage changes, control budget/schedule.

5. Closure: Formal acceptance, handover, lessons learned.

8.0 Game Theory (Introduction)

8.1 Strategic Game Concepts

• Players: Decision-makers (e.g., firm A vs. firm B).

• Strategies: Options available to each player.

• Payoffs: Numerical outcomes for each strategy combination.

• Payoff Matrix: Rows = Player 1 strategies, Columns = Player 2 strategies, entries = (Payoff to P1, Payoff to P2).

8.2 Solution Concepts

• Pure Strategies: Players choose a single deterministic strategy.

  • Saddle Point: Entry that is minimum in its row and maximum in its column.

  • Maximin-Minimax Principle: Player 1 maximizes minimum payoff; Player 2 minimizes maximum loss. Value at saddle point = game value.

• Mixed Strategies: Players choose strategies with probabilities.

  • Expected payoff = Σ (probability of P1's strategy × probability of P2's strategy × payoff).

  • Solve via linear programming or equalizing opponent's expected payoff.

• Basic Assumptions:

  1. Rational players (maximize payoff/minimize loss).

  2. Fixed payoffs (no cooperation).

  3. Simultaneous moves (or no knowledge of opponent's move).

8.3 Dominance Rule

• Dominant Strategy: Strategy $$\displaystyle S_i $$ is strictly dominant if it yields higher payoff than any other strategy regardless of opponent's choice.

• Dominated Strategy: Strategy $$\displaystyle S_i $$ is strictly dominated if another strategy yields higher payoff for all opponent's choices.

• Elimination: Remove dominated rows/columns to simplify matrix before solving.

[!TIP] Past papers (May 2024) ask to "explain pure and mixed strategies" and "dominance rule." Focus on definitions and step-by-step elimination process.

9.0 Heuristic & Meta-Heuristic Algorithms

9.1 Need for Heuristics

• For NP-hard combinatorial problems (e.g., Traveling Salesman, Vehicle Routing), exact methods (like branch-and-bound) become computationally infeasible for large instances.

• Heuristics provide good feasible solutions quickly with no guarantee of optimality.

9.2 Heuristic Algorithms

• Definition: Problem-specific "rule-of-thumb" that guides search.

• Examples:

  • Nearest Neighbor (TSP): Start at a city, repeatedly go to nearest unvisited city.

  • Savings Algorithm (VRP): Start with each customer on separate route, merge routes if saving in distance > threshold.

• Characteristics: Fast, simple, problem-dependent, may get stuck in local optimum.

9.3 Meta-Heuristic Algorithms

• Definition: High-level, problem-independent frameworks that guide heuristics to escape local optima and explore search space globally.

• General Structure:

  1. Initialization: Generate initial solution(s).

  2. Iteration: Generate new solutions via operators (mutation, crossover, perturbation), accept/reject based on criteria.

  3. Termination: After fixed iterations/time, or no improvement.

• Examples:

  • Genetic Algorithms (GA): Evolution-inspired (selection, crossover, mutation).

  • Simulated Annealing (SA): Mimics annealing process; accepts worse solutions with probability to escape local minima.

  • Tabu Search (TS): Uses memory (tabu list) to avoid revisiting recent solutions.

  • Ant Colony Optimization (ACO): Simulates ant pheromone trails for path finding.

• Advantage: More robust, better at finding near-optimal solutions for complex problems.

[!TIP] Distinguish: Heuristic = specific rule for a problem. Meta-Heuristic = general framework that can be adapted to many problems. Past papers (Jun 2025) ask for "short note" → define, give 1-2 examples, mention purpose (escape local optima).