Reliability Engineering (ME-703 (D)) - Unit 3 Short Notes

UNIT 3: OPERATIONS & SUPPLY CHAIN ANALYTICS

I. LINEAR PROGRAMMING (LP) FOUNDATIONS

A. Problem Formulation

• Process:

  1. Identify Decision Variables: Define x₁, x₂, ... representing quantities to be determined.

  2. Construct Objective Function: Express the goal (Maximize Z or Minimize Z) as a linear function of decision variables.

  3. Identify Constraints: Translate all resource limitations, demand requirements, and technical restrictions into linear inequalities/equalities.

  4. Add Non-Negativity: xᵢ ≥ 0 for all i.

• Standard Form (for Simplex): Maximization problem with ≤ constraints and RHS ≥ 0. Slack variables are added to convert inequalities to equalities.

B. Simplex Method (Maximization)

• Step-by-Step Procedure:

  1. Convert to standard form; add slack variables.

  2. Initial Tableau: Construct simplex tableau with objective row (Z row) and constraint rows. Coefficients of slack variables form an identity matrix → Basic Feasible Solution (BFS).

  3. Optimality Test: Check Z-row coefficients (net evaluation cⱼ - zⱼ). If all ≥ 0, current solution is optimal.

  4. Entering Variable: Choose most negative cⱼ - zⱼ (key column).

  5. Leaving Variable: Compute Minimum Ratio Test: (RHS / Pivot Column) for positive pivot column entries. Smallest ratio determines key row (leaving basic variable). If no positive entries → Unbounded Solution.

  6. Pivot Operation: Use Gauss-Jordan elimination to make pivot element = 1 and all other elements in pivot column = 0.

  7. Repeat from Step 3.

[!TIP] Common Pitfall: Forgetting to update the entire tableau during pivot operations. Always recalculate the Z-row using zⱼ = Σ (c_B * aᵢⱼ).

C. Special Cases & Concepts

• Degeneracy: Occurs when the minimum ratio test yields two or more equal smallest ratios. The leaving variable may become zero in the next BFS. This can lead to cycling (infinite loop) in theory, but rare in practice. Resolved by lexicographic method or artificial variables (Big-M).

• Big-M Method: Used for ≥ or = constraints. Add artificial variables with a large penalty M in the objective function to force them out of the basis.

II. TRANSPORTATION PROBLEMS (TP)

A. Initial Basic Feasible Solutions (IBFS)

1. North-West Corner Rule (NWCR)

• Steps:

  1. Start at cell (1,1) (top-left/north-west).

  2. Allocate as much as possible: xᵢⱼ = min(Supplyᵢ, Demandⱼ).

  3. Adjust supply & demand. If supply exhausted, move down; if demand satisfied, move right.

  4. Repeat until all allocations done.

• Result: Always yields a degenerate or non-degenerate BFS with exactly (m+n-1) allocations (if non-degenerate).

2. Vogel's Approximation Method (VAM)

• Steps:

  1. For each row and column, calculate penalty = difference between two smallest costs.

  2. Identify row/column with highest penalty.

  3. In that row/column, allocate to the cell with minimum cost.

  4. Adjust supply/demand, cross out satisfied row/column (recalculate penalties for remaining).

  5. Repeat until all allocations done.

• Advantage: Generally gives a better initial solution (lower cost) than NWCR.

B. Degeneracy in Transportation Problems

• Definition: When number of positive allocations < (m + n - 1) in an IBFS or during iteration.

• Resolution: Introduce an artificial allocation (ε) in an unoccupied cell with zero cost (or very small cost) to make the solution non-degenerate and proceed with the MODI method. The ε is removed after the first iteration.

C. Advanced Transportation Models

• Unbalanced TP: If Total Supply ≠ Total Demand.

  • Dummy Row/Column: Add a dummy destination with zero cost if supply > demand, or dummy origin if demand > supply.

• TP with Penalties (Unfulfilled Demand):

  • Add a dummy destination (if demand > supply) with penalty costs for unfulfilled demand from each origin.

  • The model minimizes Total Transportation Cost + Total Penalty Cost.

• Maximization TP: Convert to minimization by subtracting all costs from a large constant (e.g., max cost in matrix + 1).

III. INVENTORY MANAGEMENT (EOQ MODELS)

A. Basic Economic Order Quantity (EOQ) Model

• Assumptions:

  • Demand rate D is constant & known.

  • Instantaneous replenishment (order arrives all at once).

  • No shortages allowed.

  • Fixed ordering cost S per order.

  • Constant holding cost H per unit per year.

• Derivation: Minimize Total Annual Cost (TAC).

  • Number of orders/year = D / Q

  • Ordering Cost/year = (D/Q) * S

  • Avg. Inventory = Q/2

  • Holding Cost/year = (Q/2) * H

  • TAC(Q) = Purchase Cost + (D S / Q) + (H Q / 2)

• Optimal Order Quantity (EOQ):

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

• Optimal Total Annual Cost (excluding purchase cost):

$$\boxed{TAC^* = \sqrt{2DSH}}$$

B. Key Parameters from EOQ

• Optimum Number of Orders per Year: N = D / Q*

• Optimum Time Between Orders (Cycle Time): T = 365 / N days or T = Q* / D years.

• Reorder Point (ROP): ROP = d * L (where d = daily demand, L = lead time). Assumes no safety stock.

C. Extensions & Sensitivity

• EOQ with Finite Production Rate (P): Production rate > demand rate (P > d). Inventory builds gradually.

$$\boxed{Q^* = \sqrt{\frac{2DS}{H \left(1 - \frac{d}{P}\right)}}}$$

• EOQ with Quantity Discounts:

  • All-Units Discount: Discount applies to all units if order quantity ≥ discount breakpoint.

    • Compute EOQ at each price (using H = i * C, where i = carrying cost %).

    • If EOQ is below breakpoint, evaluate TAC at breakpoint quantity.

    • Choose quantity with lowest TAC.

  • Incremental Discount: Discount applies only to units above breakpoint.

    • Compute EOQ at lowest price first. If feasible, it's optimal. If not, move to next price level.

• Sensitivity: Q* ∝ √D, √S, 1/√H. A 25% change in D, S, H changes Q* by ~12.5%.

IV. SUPPLY CHAIN MANAGEMENT (SCM) CORE CONCEPTS

A. SCM Framework & Flows

• Three Key Flows:

  1. Material Flow: Physical movement of goods from suppliers to customers.

  2. Information Flow: Transmission of orders, forecasts, and status updates (bidirectional).

  3. Money Flow (Financial Flow): Movement of funds, credit, and payments (from customer to supplier).

• Objective: To achieve net value creation, maximize total surplus, and deliver value to end customer efficiently.

B. Key SCM Functions

• Inbound Logistics: Activities related to incoming materials from suppliers. Includes sourcing, purchasing, transportation, receiving, and inventory management of raw materials. Goal: Ensure right materials, right quantity, right time, right cost.

• Outbound Logistics: Activities related to finished goods moving to customers. Includes order processing, warehousing, transportation, and distribution. Goal: Meet customer service levels at minimum cost.

• Cross-Docking:

  • Process: Inbound trucks unload directly to outbound trucks with minimal storage (< 24 hrs). Goods are sorted and consolidated.

  • Benefits: Reduces inventory holding, handling, storage space, and order cycle time.

  • Disadvantages: Requires high coordination, accurate forecasting, and synchronized transportation. Not suitable for all products.

C. SCM Phenomena & Strategies

• Bull-Whip Effect: Amplification of demand variability as one moves upstream in the supply chain (from retailer to manufacturer).

  • Causes:

    1. Demand Forecast Updating (using orders, not sales).

    2. Order Batching (periodic ordering).

    3. Price Fluctuations (forward buying).

    4. Rationing & Gaming (shortage-induced over-ordering).

  • Mitigation Strategies:

    1. Vendor Managed Inventory (VMI): Supplier manages inventory at customer location.

    2. Continuous Replenishment: Share POS data, base orders on actual sales.

    3. Reduce Lead Times.

    4. Stabilize Prices.

    5. Eliminate order batching through EDI/Internet.

• Outsourcing: Contracting non-core activities (e.g., logistics, IT, manufacturing) to third-party specialists (3PL/4PL). Strategic Importance: Focus on core competencies, reduce costs, gain flexibility, access expertise.

• Expenditure & Opportunities: SCM is a major source of cost reduction (logistics ~10-15% of GDP) and competitive advantage (speed, reliability, responsiveness).

D. Evolution & Integration

• MRP → ERP → SCM:

  • MRP (Material Requirements Planning): Focuses on internal material planning and scheduling (dependent demand). Competitive Advantages: Reduced inventory, better production scheduling, improved capacity utilization.

  • ERP (Enterprise Resource Planning): Integrates all internal functions (finance, HR, manufacturing, sales) into a single system.

  • SCM: Extends integration beyond the firm to include suppliers, distributors, and customers. Focus on external coordination and total pipeline optimization.

• Linking SCM with E-Business: E-commerce platforms (B2B, B2C) enable real-time information sharing, electronic ordering, and collaborative planning (CPFR), reducing transaction costs and improving visibility.

V. PROJECT MANAGEMENT (PERT/CPM)

A. Fundamentals & Comparison

B. Network Construction & Analysis

• Network Logic Rules (AOA):

  1. Each activity represented by an arrow.

  2. Events (nodes) mark start/end of activities.

  3. Dummy Activity (zero duration) used to resolve logic/dependencies.

  4. No crossing of arrows if possible.

• Forward Pass (Calculate ES/EF):

  • ES(i) = 0 for start node.

  • EF(i) = ES(i) + t(i)

  • ES(j) = max[EF(i)] for all immediate predecessor activities i of j.

• Backward Pass (Calculate LS/LF):

  • LF(n) = EF(n) for final node (project duration).

  • LS(i) = LF(i) - t(i)

  • LF(i) = min[LS(j)] for all immediate successor activities j of i.

• Slack/Float: Total Float = LS - ES = LF - EF. Critical Path: Path with zero total float (longest path through network).

C. PERT Time Estimates & Project Duration

• Three-Time Estimates:

  • tₒ = Optimistic time (best case, everything goes right).

  • tₘ = Most likely time (most realistic).

  • tₚ = Pessimistic time (worst case, everything goes wrong).

• Expected Activity Time:

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

• Activity Variance:

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

• Project Duration & Variance:

  • Expected Project Duration (Tₑ): Sum of tₑ on the Critical Path.

  • Project Variance (σₚ²): Sum of σ² on the Critical Path (assuming independence).

D. Probability Analysis (PERT)

• Assumption: Project completion time follows a Normal Distribution with mean Tₑ and variance σₚ².

• Z-Score Calculation:

$$Z = \frac{\text{Due Date} - T_e}{\sqrt{\sigma_p^2}}$$

• Probability: Use standard normal table to find P(Z ≤ z). Probability of completion by due date = P(Z ≤ z).

• **Probability of completion after due date = 1 - P(Z ≤ z).

[!TIP] Exam Focus: Always identify the critical path first. Project variance is only the sum of variances on the critical path.

VI. QUEUING THEORY (BASIC)

A. Core Concepts

• Queue: Customers waiting in line for service.

• Arrival Rate (λ): Mean number of customers arriving per unit time. Assumed Poisson → inter-arrival times exponential.

• Service Rate (μ): Mean number of customers served per unit time by one server. Assumed Exponential.

• Utilization Factor (ρ): ρ = λ / μ. Must be < 1 for a steady-state system.

• Queue Disciplines: Rules for selecting next customer for service.

  • FIFO/FCFS: First-In-First-Come-First-Served (most common).

  • LIFO/LCFS: Last-In-First-Come-First-Served (stack).

  • SIRO: Service In Random Order.

  • Priority: Based on customer class/urgency.

  • Random: Any customer selected randomly.

B. Key Distributions

• Poisson Distribution (Arrivals):

  • Probability of k arrivals in time t: P(k) = (e^(-λt) * (λt)^k) / k!

• Exponential Distribution (Service Times):

  • PDF: f(t) = μ e^(-μt) for t ≥ 0.

  • CDF: F(t) = 1 - e^(-μt).

  • Memoryless Property: P(T > s + t | T > s) = P(T > t).

  • Mean Service Time: 1/μ.

C. Performance Measures (Exponential Service, Single Server, Infinite Capacity - M/M/1)

• Probability that waiting time Wq exceeds t:

$$\boxed{P(W_q > t) = e^{-(μ - λ)t}}$$

*   **Interpretation:** Probability a customer waits more than `t` time **before service begins**.

• Other Key Measures (Conceptual):

  • Lq = Avg. number in queue = (λ²) / (μ(μ-λ))

  • Ls = Avg. number in system = λ / (μ-λ)

  • Wq = Avg. waiting time in queue = λ / (μ(μ-λ))

  • Ws = Avg. time in system = 1 / (μ-λ)

[!TIP] Exam Focus: The formula P(T > t) = e^(-μt) is for service time distribution only. For waiting time in M/M/1 queue, use P(Wq > t) = e^{-(μ-λ)t}. Always check if the question asks about service time or waiting time.

VII. GAME THEORY (BASIC)

A. Strategies & Assumptions

• Pure Strategy: A player chooses a specific action with certainty.

• Mixed Strategy: A player randomizes over actions according to a probability distribution.

• Basic Assumptions:

  1. Finite number of players (usually 2 for basic games).

  2. Finite set of strategies for each player.

  3. Players are rational (seek to maximize their own payoff).

  4. Payoffs are known to all players (perfect information about the game matrix).

  5. Decisions are made simultaneously or without knowledge of the other's choice.

B. Solution Concepts

• Dominance Rule:

  • Row Dominance: Row i dominates row j if aᵢₖ ≥ aⱼₖ for all columns k and > for at least one k. Dominated row j can be deleted.

  • Column Dominance: Column i dominates column j if aₖᵢ ≤ aₖⱼ for all rows k and < for at least one k. Dominated column j can be deleted.

  • Iterative Elimination: Repeatedly remove dominated rows/columns to reduce game size.

VIII. INVENTORY CLASSIFICATION & ANALYSIS

A. ABC Analysis (Pareto Analysis)

• Principle: 80/20 rule – ~80% of inventory value comes from ~20% of items.

• Procedure:

  1. Compute Annual Usage Value (AUV) for each item: AUV = Annual Demand × Unit Cost.

  2. List items in descending order of AUV.

  3. Calculate cumulative AUV and cumulative percentage.

  4. Classify:

     • A-items: Top ~70-80% of cumulative AUV (usually ~10-20% of items). Tight control, frequent review, accurate records.

     • B-items: Next ~15-25% of cumulative AUV (~30% of items). Normal control, periodic review.

     • C-items: Remaining ~5% of cumulative AUV (~50% of items). Simple controls, large inventories, minimal record-keeping.

• Purpose: Prioritize management effort and capital where it matters most.

B. VED Analysis (Vital, Essential, Desirable)

• Basis: Criticality of items to the production system or service, not monetary value.

  • V (Vital): Items whose shortage stops production completely (e.g., key spare parts, unique raw materials). Highest priority, tightest control, maximum safety stock.

  • E (Essential): Items whose shortage seriously affects production but doesn't stop it entirely. High priority, good control.

  • D (Desirable): Items whose shortage causes minor inconvenience. Lowest priority, simple controls, minimal stock.

• Application: Often used for spare parts inventory in maintenance.

C. Advantages of ABC & VED

• Focused Management: Resources (capital, attention) directed to high-impact items.

• Optimized Inventory Costs: Reduces capital tied up in C-items, prevents stockouts of A/V-items.

• Improved Resource Allocation: Efficient use of storage space and administrative effort.

• Better Control: Different control policies (review frequency, safety stock levels) applied appropriately.

• Enhanced Decision-Making: Clear prioritization for purchasing, storage, and stock replenishment.

IX. HEURISTIC & META-HEURISTIC ALGORITHMS (CONCEPTUAL)

• Need: For NP-Hard problems (e.g., TSP, VRP, Facility Location) where exact methods (like Simplex) become computationally intractable for large instances. Heuristics provide "good enough" solutions quickly.

• Heuristic: A problem-specific rule-of-thumb or procedure that guides the search for a solution. Not guaranteed to be optimal, but often fast and effective.

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

  • Example (VRP): Savings Algorithm (Clarke-Wright) – Start with each customer on a separate route, then merge routes based on distance savings.

• Meta-Heuristic: A higher-level, problem-independent framework that guides the search process. Can be applied to a wide range of problems.

  • Examples: Genetic Algorithms (evolutionary), Simulated Annealing (thermal process analogy), Tabu Search (memory-based), Ant Colony Optimization.

  • Key Feature: Incorporates mechanisms to escape local optima (e.g., mutation in GA, probabilistic acceptance in SA) to explore the solution space more broadly.

• Distinction from Exact Methods: Exact methods (e.g., Branch & Bound) guarantee optimality but may take exponential time. Heuristics/Meta-heuristics trade optimality for speed on large-scale problems.