UNIT 2: OPERATION RESEARCH & SUPPLY CHAIN - EXAM-FOCUSED NOTES

I. LINEAR PROGRAMMING (LP)

A. Problem Formulation & Mathematical Modeling

• Process: Identify Decision Variables (what to control), Objective Function (Maximize/Minimize), and Constraints (limitations).

• Standard Form Conversion:

  • Maximization: Max Z = cᵀx

  • Constraints: All ≤ converted using slack variables (+s), all ≥ converted using surplus variables (-s) + artificial variables (+a) for initial feasible solution.

  • Non-negativity: xᵢ ≥ 0 for all i.

• Example Formulation Structure:

  Let x₁, x₂ = units of product A, B produced.

  Max Z = p₁x₁ + p₂x₂ (Profit)

  s.t.

  a₁₁x₁ + a₁₂x₂ ≤ b₁ (Resource 1)

  a₂₁x₁ + a₂₂x₂ ≤ b₂ (Resource 2)

  x₁, x₂ ≥ 0

B. Simplex Method (Primary Solution Technique)

• Objective: Move from one Basic Feasible Solution (BFS) to a better BFS along edges of feasible region until optimal.

• Key Tableau Components:

  • Cj: Objective coefficients.

  • Cj - Zj: Net Evaluation Row. For maximization, optimal when all Cj - Zj ≤ 0.

  • Pivot Column: Most positive Cj - Zj (entering variable).

  • Pivot Row: Minimum positive ratio (RHS / Pivot Column entry) (leaving variable).

• Step-by-Step:

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

  2. Construct initial simplex tableau.

  3. Identify entering variable (most positive Cj - Zj).

  4. Identify leaving variable (min positive ratio test).

  5. Perform pivot operation (Gauss-Jordan elimination) to make pivot element = 1, others in column = 0.

  6. Repeat from step 3 until optimality condition met.

• Special Cases:

  • Unbounded Solution: All entries in pivot column ≤ 0 → objective can increase indefinitely.

  • Multiple Optimal Solutions: Cj - Zj = 0 for a non-basic variable in optimal tableau.

  • Degeneracy: Basic variable becomes zero in a solution. Can cause cycling. Resolution: Use artificial variables (Big-M or Two-Phase method) or perturbation method.

[!TIP] Exam Focus: Every paper has a 14-mark simplex problem. Practice full iterations, including handling artificial variables (Big-M/Two-Phase). Always check optimality condition correctly (≤0 for max, ≥0 for min).

C. Sensitivity Analysis & Duality (Implied)

• Shadow Price (Dual Value): Change in objective per unit increase in RHS of a constraint, ceteris paribus. Indicates resource scarcity value.

• Allowable Ranges: Range over which current optimal basis remains optimal.

  • For RHS bᵢ: bᵢ_new must satisfy bᵢ_min ≤ bᵢ_new ≤ bᵢ_max.

  • For objective coeff cⱼ: cⱼ_new must satisfy cⱼ_min ≤ cⱼ_new ≤ cⱼ_max.

II. TRANSPORTATION PROBLEMS (TP)

A. Problem Structure

• Balanced TP: Total Supply = Total Demand.

• Unbalanced TP: Add Dummy Row (if supply < demand) or Dummy Column (if supply > demand) with zero transportation cost.

• Objective: Minimize total transportation cost.

B. Finding Initial Basic Feasible Solution (IBFS)

C. Optimality Test & Iteration (MODI / UV Method)

• Step 1: For IBFS with m+n-1 allocations, find uᵢ, vⱼ such that uᵢ + vⱼ = Cᵢⱼ for all basic cells.

  • Set u₁ = 0 (or any arbitrary value), solve for others.

• Step 2: Calculate opportunity cost for all non-basic cells: Δᵢⱼ = Cᵢⱼ - (uᵢ + vⱼ).

• Step 3: Optimality Check:

  • Minimization: Optimal if all Δᵢⱼ ≥ 0.

  • If any Δᵢⱼ < 0, solution is non-optimal.

• Step 4 (Improvement): Select most negative Δᵢⱼ. Form closed loop connecting cells with allocations. Adjust allocations: +θ at (i,j), -θ alternately along loop. New solution is IBFS.

D. Degeneracy in Transportation

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

• Cause: Simultaneous exhaustion of a row and column during allocation.

• Resolution:

  1. Assign a very small quantity ε (e.g., 0.001) to one or more zero-allocation cells to make total allocations = m+n-1.

  2. Proceed with MODI/Stepping Stone method treating ε as a real allocation.

  3. Final solution will not include ε.

[!TIP] Exam Focus: VAM for IBFS is heavily tested. MODI method for optimality is mandatory. Be prepared to explain degeneracy and its resolution (Jun 25).

E. Advanced TP Models

• Transshipment: Intermediate nodes where goods can be transferred. Solved by converting to standard TP with dummy origins/destinations or using LP directly.

• TP with Penalties (Unfulfilled Demand): Add dummy destination (if demand > supply) with penalty cost Pⱼ for each original destination j. Minimize Total Cost + Total Penalty.

• Maximization TP: Convert to minimization by subtracting all costs from a large constant M (e.g., M = max(Cᵢⱼ) + 1). Solve as min. problem. Optimal allocation remains same.

III. INVENTORY MANAGEMENT

A. Economic Order Quantity (EOQ) Model - Cornerstone

• Assumptions:

  1. Demand D is known, constant, deterministic.

  2. Replenishment is instantaneous (delivery in one batch).

  3. No shortages allowed.

  4. Ordering cost S per order is fixed.

  5. Holding/carrying cost H per unit per year is constant.

• Derivation: Minimize Total Cost (TC) = Purchase Cost + Ordering Cost + Holding Cost.

  TC = D*C + (D/Q)*S + (Q/2)*H

• Optimal Order Quantity:

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

• Key Results:

  • Number of orders/year: N = D / Q*

  • Time between orders: T = Q* / D (years)

  • Minimum Total Cost (variable): TC_min = \sqrt{2DSH}

  • Maximum Inventory Level: Q*

  • Average Inventory: Q*/2

B. EOQ Variations

1. Finite Production Rate (EPQ / Production Model):

   • Production rate P > demand rate D.

   • Maximum inventory builds gradually: I_max = Q*(1 - D/P).

$$\boxed{EPQ = Q^* = \sqrt{\frac{2DS}{H(1 - D/P)}}}$$

2. Quantity Discounts:

   • Procedure:

     1. Compute EOQ using H = i*C (where i = carrying cost %).

     2. If EOQ falls within a price break's range, calculate TC at EOQ.

     3. If EOQ is below a price break's minimum quantity, compute TC at that minimum quantity (using its price).

     4. Compare TCs for all feasible price breaks. Choose minimum.

   • All-units discount: Entire order gets discount price if quantity ≥ break.

   • Incremental discount: Only units above break get discounted price.

3. Shortages Allowed (Backordering):

   • Shortage cost B per unit per year.

   • Optimal order quantity:

$$\boxed{Q^* = \sqrt{\frac{2DS}{H} \cdot \frac{H+B}{B}}}$$

*   Maximum backorder level: `B* = Q* * [B / (H+B)]`

*   Average inventory: `(Q* - B*)/2`

C. Inventory Classification

• ABC Analysis (Pareto Principle):

  • Classify items based on Annual Usage Value = Annual Demand (D) × Unit Cost (C).

  • A-class: ~70-80% of total value, ~10-20% of items. Tight control.

  • B-class: ~15-25% of value, ~20-30% of items. Normal control.

  • C-class: ~5-10% of value, ~50-70% of items. Loose control.

• VED Analysis (For Spare Parts):

  • V (Vital): No stock = production stops. Highest priority.

  • E (Essential): High stock needed, but alternatives may exist.

  • D (Desirable): Low impact on operations. Can be stocked less.

• Advantages of ABC/VED: Focuses control efforts, reduces inventory costs, improves resource allocation.

[!TIP] Exam Focus: EOQ (basic & variations) appears every paper. Be ready to calculate EOQ, TC, N, T. Discount analysis (Nov 23) and ABC/VED explanation (Jun 25) are high-probability.

IV. QUEUEING THEORY

A. Fundamental Concepts & Kendall's Notation

• Notation: A/B/c

  • A: Arrival process distribution (e.g., M = Markovian/Poisson).

  • B: Service time distribution (e.g., M = Exponential).

  • c: Number of service channels.

• Key Parameters:

  • λ = Mean arrival rate (customers/unit time).

  • μ = Mean service rate (customers/unit time per channel).

  • ρ = λ / (cμ) = Utilization factor (must be < 1 for stable system).

B. Single-Server Queue (M/M/1) - Primary Focus

• Performance Measures (Steady-State):

$$P_n = (1 - \rho) \rho^n \quad \text{(Prob. of n customers in system)}$$

$$L = \frac{\rho}{1 - \rho} \quad \text{(Avg. number in system)}$$

$$L_q = \frac{\rho^2}{1 - \rho} \quad \text{(Avg. number in queue)}$$

$$W = \frac{1}{\mu - \lambda} \quad \text{(Avg. time in system)}$$

$$W_q = \frac{\lambda}{\mu(\mu - \lambda)} \quad \text{(Avg. waiting time in queue)}$$

• Probability of waiting time t:

$$P(T > t) = e^{-\mu(1-\rho)t} \quad \text{(for exponential service times)}$$

• Idle time probability: P₀ = 1 - ρ.

C. Multi-Server Queue (M/M/c) - Conceptual

• Stability condition: ρ = λ/(cμ) < 1.

• More complex formulas for P₀, L_q, etc. (usually not derived in exam).

• Key Insight: Adding servers reduces L_q and W_q significantly once ρ is high.

[!TIP] Exam Focus: Probability calculations for M/M/1 are frequent (Jun 25, Dec 24, May 24, Nov 23). Remember: P(T > t) = e^{-μ(1-ρ)t}. Convert units consistently (e.g., minutes to hours).

V. PROJECT MANAGEMENT (PERT/CPM)

A. Network Diagram Construction (AON/PDM)

• Activity-on-Node (AON): Box = activity, Arrow = dependency (finish-to-start most common).

• Activity-on-Arrow (AOA): Arrow = activity, Node = event. Requires dummy activities (dashed arrows, zero time) to show dependencies when two activities share same start/end events.

• Rules: No crossing arrows if possible; maintain left-to-right flow.

B. Critical Path Method (CPM)

• Forward Pass (Earliest Times):

  • ESᵢ = Earliest Start of activity i.

  • EFᵢ = ESᵢ + tᵢ

  • For successor j: ESⱼ = max(EFᵢ) over all predecessors i.

• Backward Pass (Latest Times):

  • LFᵢ = Latest Finish of activity i without delaying project.

  • LSᵢ = LFᵢ - tᵢ

  • For predecessor i: LFᵢ = min(LSⱼ) over all successors j.

• Float/Slack:

  • Total Float (TF): TFᵢ = LSᵢ - ESᵢ = LFᵢ - EFᵢ. Time an activity can be delayed without delaying project.

  • Free Float (FF): Delay without delaying early start of successor.

• Critical Path: Path with zero Total Float. Longest path through network. Determines project duration.

C. Program Evaluation and Review Technique (PERT)

• Three-Time Estimates:

  • a = Optimistic time (best case).

  • m = Most likely time.

  • b = Pessimistic time (worst case).

• Expected Time & Variance:

$$\boxed{t_e = \frac{a + 4m + b}{6}}$$

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

• Project Completion Probability (Normal Distribution):

  • Project expected duration Tₑ = Σ tₑ on critical path.

  • Project variance σₚ² = Σ σᵢ² on critical path.

  • For due date T_d, Z = (T_d - Tₑ) / σₚ. Use standard normal table.

D. PERT vs. CPM

[!TIP] Exam Focus: Drawing network and finding critical path (Jun 25). PERT probability calculation (Nov 23). Comparison question (Dec 24, May 24). Remember: t_e and σ² formulas are boxed.

VI. SUPPLY CHAIN MANAGEMENT (SCM)

A. Core Flows in SCM

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

2. Information Flow: Orders, forecasts, shipment notifications, invoices.

3. Money/Cash Flow: Credit terms, payments, consignments, investments.

B. Logistics in SCM

• Inbound Logistics: Activities from supplier receipt to internal warehouse. Includes sourcing, transportation, receiving, inventory management. Goal: Efficient, cost-effective material inflow.

• Outbound Logistics: Activities from warehouse to customer. Includes order processing, warehousing, transportation, delivery, reverse logistics (returns, recycling). Goal: Timely, accurate, cost-effective customer delivery.

C. Bull-Whip Effect

• Definition: Demand information distortion as it moves upstream (retailer → wholesaler → distributor → manufacturer). Small demand fluctuations at consumer level cause large order fluctuations at manufacturer.

• Causes:

  1. Demand forecast updating (each tier forecasts based on orders, not consumer demand).

  2. Order batching (periodic large orders).

  3. Price fluctuations (forward buying).

  4. Rationing & gaming (supply shortage → over-ordering).

• Mitigation Strategies:

  • Vendor Managed Inventory (VMI): Supplier manages inventory at customer's location.

  • Continuous Replenishment: Share POS data, frequent small deliveries.

  • Eliminate incentives: Avoid forward buying, quantity discounts.

  • Information Sharing: Point-of-Sale (POS) data across chain.

  • Order smoothing: Stabilize ordering patterns.

D. Key SCM Strategies & Technologies

• Cross-Docking:

  • Definition: Inbound shipments are directly transferred to outbound trucks with minimal/zero storage.

  • Importance: Reduces inventory holding, handling, storage costs; speeds up flow.

  • Disadvantages: Requires excellent coordination, real-time information systems, reliable transportation, high initial investment.

• Outsourcing / 3PL (Third-Party Logistics):

  • Importance: Focus on core competencies, access to expertise/technology, cost savings, scalability, risk sharing.

  • Risks: Loss of control, dependency, hidden costs, quality issues.

• MRP → MRP II → ERP Evolution:

  • MRP (Material Requirements Planning): Explosion of BOM using MPS, inventory records, lead times. Focus: Materials.

  • MRP II (Manufacturing Resource Planning): MRP + capacity planning, shop floor control, finance. Focus: Manufacturing resources.

  • ERP (Enterprise Resource Planning): MRP II + integrated modules (HR, CRM, SCM, finance) across entire enterprise. Focus: Enterprise-wide integration.

  • Link to SCM: ERP provides the information backbone for SCM, enabling visibility, coordination, and planning across the supply chain.

• Role of Inventory in Logistics:

  • Buffer: Against demand/supply uncertainty.

  • Economies of Scale: Larger orders reduce per-unit costs.

  • Decoupling Point: Separates push (forecast-driven) from pull (demand-driven) strategies.

  • Trade-offs: Holding cost vs. Stock-out cost vs. Ordering cost vs. Responsiveness.

[!TIP] Exam Focus: Bull-whip effect (causes & mitigation) is in every paper. Cross-docking (importance & disadvantages) (Nov 23). Inbound/Outbound logistics (Dec 24, May 24). MRP to ERP evolution (Jun 25). Role of inventory (Jun 25). Flow of material, money, information (Dec 24).

VII. ADVANCED TOPICS

A. Game Theory (Basic)

• Assumptions: Rational players, known payoffs, simultaneous moves (usually), single play or repeated.

• Pure Strategy: Specific choice (row/column) selected.

• Mixed Strategy: Probability distribution over pure strategies.

• Dominance Rule:

  • Row Dominance: If aᵢⱼ ≥ aₖⱼ for all j and > for at least one j, then row i dominates row k. Delete dominated row.

  • Column Dominance: If aᵢⱼ ≤ aᵢₖ for all i and < for at least one i, then column j dominates column k. Delete dominated column.

  • Purpose: Reduce payoff matrix size before solving.

B. Heuristic & Metaheuristic Algorithms

• Heuristics: Problem-specific, fast, simple rules of thumb. Provide good (not necessarily optimal) solutions quickly. Example: Nearest Neighbor for TSP.

• Metaheuristics: General-purpose, high-level frameworks that guide heuristics to escape local optima. Examples: Genetic Algorithms (evolution), Simulated Annealing (cooling process), Tabu Search (memory-based).

• Purpose: Solve large, complex, NP-hard problems where exact methods (like simplex) are computationally infeasible.

C. Network Logics (AON)

• Dependency Types (FS, SS, FF, SF):

  • Finish-to-Start (FS): Successor cannot start until predecessor finishes. Most common.

  • Start-to-Start (SS): Successor cannot start until predecessor starts.

  • Finish-to-Finish (FF): Successor cannot finish until predecessor finishes.

  • Start-to-Finish (SF): Successor cannot finish until predecessor starts. Rare.

• Lag/Lead: Time offsets added to dependencies (e.g., FS+5 means start 5 days after predecessor finishes).

[!TIP] Exam Focus: Short notes on Heuristics/Metaheuristics, Network Logics (Jun 25). Pure/Mixed strategies & assumptions (May 24). Dominance rule (Dec 24).

FINAL EXAM STRATEGY:

1. Tier 1 Problems (14m): Simplex, Transportation (VAM+MODI). Practice 3-4 full problems each. Master tableau setup, pivot operations, degeneracy handling.

2. Theory & Application (7m): Bull-whip, Logistics (Inbound/Outbound), EOQ variations, PERT/CPM comparison. Use bullet points, diagrams (network), formulas.

3. Numerical Probability (7m): Queueing (M/M/1), PERT completion probability. Write formula, substitute values, state final probability.

4. Definitions & Comparisons (4-7m): ABC/VED, Cross-docking, MRP/ERP, Game theory terms, Network logics. Be concise, structured.

5. Always: Box final formulas. Show clear steps in numerical problems. Interpret results (e.g., shadow price, critical path).