UNIT 5: OPERATIONS RESEARCH & SUPPLY CHAIN MANAGEMENT

1.0 LINEAR PROGRAMMING (LP)

1.1 Formulation of LP Problems

• Decision Variables: Quantities to be determined (e.g., \(x_1, x_2\)).

• Objective Function: Mathematical expression to be maximized (profit, revenue) or minimized (cost, time).

  • General form: Maximize/Minimize \(Z = c_1x_1 + c_2x_2 + ... + c_nx_n\)

• Constraints: Limitations expressed as linear inequalities/equalities (resource capacities, demand, technology).

  • Example: \(a_{11}x_1 + a_{12}x_2 \le b_1\)

• Non-negativity Restrictions: \(x_j \ge 0\) for all \(j\).

• Standard Form Conversion (for Simplex):

  • Maximization objective.

  • All constraints as equalities using:

    • Slack Variables (≤ constraints): Add \(s \ge 0\).

    • Surplus Variables (≥ constraints): Subtract \(s \ge 0\).

    • Artificial Variables (equality or ≥ constraints): Add \(a \ge 0\) (to be removed later via Big-M or Two-Phase method).

  • RHS (\(b_i\)) must be \(\ge 0\). If not, multiply constraint by -1.

[!TIP] Exam Focus: LP formulation is a very common 14-mark question. Always define variables clearly, state objective, list all constraints with units, and include non-negativity.

1.2 Simplex Method

Iterative Procedure for Optimal Solution:

1. Convert LP to standard form.

2. Construct initial simplex tableau using slack/artificial variables.

3. Optimality Test:

   • For maximization: If all coefficients in \(Z_j - C_j\) row are ≤ 0, current solution is optimal.

   • For minimization: If all coefficients in \(Z_j - C_j\) row are ≥ 0, current solution is optimal.

4. If not optimal, select incoming variable (most positive coefficient for max, most negative for min in \(Z_j - C_j\) row).

5. Select outgoing variable using Minimum Ratio Test (RHS / pivot column element, only for positive pivot column entries).

6. Perform pivot operation to update tableau.

7. Repeat steps 3-6 until optimality condition met.

Multiple/Optimal Solutions:

• Occurs when a non-basic variable has \(Z_j - C_j = 0\) in optimal tableau.

• Indicates alternative optimal solutions along an edge of the feasible region.

• To find another optimal solution, bring that variable into the basis and perform one iteration.

[!TIP] Exam Focus: Direct simplex problems appear very frequently. Master tableau setup, pivot operations, and optimality interpretation. Remember: For maximization, optimal when all \(Z_j - C_j \le 0\).

2.0 TRANSPORTATION & ASSIGNMENT PROBLEMS

2.1 Transportation Problem (TP)

Model Structure:

• Origins (Sources): \(m\) factories/supplies with capacities \(a_i\).

• Destinations (Markets): \(n\) warehouses/demands with requirements \(b_j\).

• Unit Transportation Cost: \(c_{ij}\) from origin \(i\) to destination \(j\).

• Objective: Minimize total transportation cost: \(\min Z = \sum_{i=1}^{m} \sum_{j=1}^{n} c_{ij} x_{ij}\)

• Constraints:

  • \(\sum_{j=1}^{n} x_{ij} = a_i\) (Supply constraints)

  • \(\sum_{i=1}^{m} x_{ij} = b_j\) (Demand constraints)

  • \(x_{ij} \ge 0\)

Initial Basic Feasible Solution (IBFS):

1. North-West Corner Rule (NWCR): Start at top-left cell (1,1). Allocate as much as possible (min of supply/demand). Move right if demand exhausted, down if supply exhausted. Repeat.

2. Vogel's Approximation Method (VAM) - \boxed{Very High Priority}

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

   • Select row/column with highest penalty.

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

   • Adjust supply/demand, cross out exhausted row/column, recalculate penalties.

   • Repeat until all allocations made.

   • Generally gives solution closer to optimal than NWCR/Least Cost.

Optimality Test: MODI / u-v Method:

1. For IBFS with \(m+n-1\) allocations, find dual variables \(u_i, v_j\) such that \(u_i + v_j = c_{ij}\) for all basic cells.

2. Compute improvement indices for non-basic cells: \(\Delta_{ij} = c_{ij} - (u_i + v_j)\).

3. Optimality: If all \(\Delta_{ij} \ge 0\), solution is optimal.

4. If any \(\Delta_{ij} < 0\), select most negative \(\Delta_{ij}\) cell for loop formation to get new solution.

Degeneracy in TP:

• Definition: Occurrence of fewer than \(m+n-1\) positive allocations in a basic feasible solution.

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

• Resolution: Introduce a dummy allocation (very small \(\epsilon > 0\)) in a zero cell to make total allocations \(m+n-1\). This cell is treated as basic in MODI calculations but its allocation is ignored for cost.

Unbalanced TP:

• If \(\sum a_i > \sum b_j\): Add dummy destination with zero cost, demand = \(\sum a_i - \sum b_j\).

• If \(\sum a_i < \sum b_j\): Add dummy origin with zero cost, supply = \(\sum b_j - \sum a_i\).

TP with Penalties/Shortages:

• Add dummy destination (if shortage allowed) with penalty cost for unfulfilled demand.

• The cost in dummy column represents penalty for not meeting that destination's requirement.

• Solve modified TP; dummy allocations indicate actual shortages.

[!TIP] Exam Focus: VAM for IBFS and MODI for optimality are very high priority. Degeneracy resolution (epsilon method) is high priority. TP with penalties appeared explicitly in a recent paper.

2.2 Assignment Problem

• Special case of TP where \(m = n\) and each origin must be assigned to exactly one destination.

• Hungarian Method:

  1. Row reduction: Subtract min of each row from all elements in that row.

  2. Column reduction: Subtract min of each column from all elements in that column.

  3. Cover all zeros with minimum number of lines (horizontal/vertical). If lines = \(n\), optimal assignment exists.

  4. If not, adjust matrix: Find smallest uncovered element, subtract it from all uncovered cells, add it to cells at intersection of lines. Repeat step 3.

• Assignment made to zero cells such that each row/column has exactly one assignment.

3.0 SUPPLY CHAIN MANAGEMENT (SCM) FUNDAMENTALS

3.1 Core SCM Concepts

• Definition: SCM encompasses all activities involved in delivering a product/service from supplier to customer, including planning, sourcing, manufacturing, delivery, and return.

• Objectives: Reduce costs, improve quality, increase speed/flexibility, enhance customer satisfaction.

• Flows in SCM:

  • Material Flow: Physical movement of goods from raw materials to end user.

  • Money Flow: Financial transactions (payments, credit, investment).

  • Information Flow: \boxed{Most Critical} Data on orders, inventory, demand forecasts, shipments. Enables coordination.

[!TIP] Exam Focus: Information flow's role in SCM coordination is a high priority theory question.

3.2 Key SCM Functions & Logistics

• Inbound Logistics: Activities related to receiving, storing, and distributing incoming materials.

  • Activities: Sourcing/procurement, inbound transportation, receiving, inventory management of raw materials.

  • Importance: Ensures smooth production by having right materials at right time/quality/cost.

• Outbound Logistics: Activities related to storing and distributing finished goods to customers.

  • Activities: Finished goods warehousing, order processing, outbound transportation, delivery.

  • Importance: Directly impacts customer service level and satisfaction.

• Role of Inventory in Logistics System:

  • Buffer Function: Decouples supply and demand uncertainties.

  • Cost Trade-offs: Holding cost vs. ordering/stockout costs.

  • Service Level: Higher inventory generally improves product availability but increases holding cost.

[!TIP] Exam Focus: Inbound vs. Outbound logistics definitions and activities are very high priority. Often asked as separate 7-mark questions.

3.3 Strategic SCM Issues

• Bull-Whip Effect:

  • Definition: Amplification of demand variability as orders move upstream (from retailer to manufacturer).

  • Causes:

    1. Demand Forecast Updating.

    2. Order Batching.

    3. Price Fluctuations.

    4. Rationing & Gaming.

  • Consequences: Excess inventory, poor capacity utilization, increased costs, poor customer service.

  • Mitigation Strategies:

    • Reduce information delays (sharing POS data).

    • Vendor Managed Inventory (VMI).

    • Stabilize prices.

    • Reduce order batching (E-commerce, EDI).

    • Eliminate gaming (allocation rules).

• Cross-Docking:

  • Definition: Logistics practice where incoming goods from suppliers are directly transferred to outbound trucks with minimal or no storage.

  • Process: Unload → Sort → Consolidate → Load.

  • Advantages: Reduced inventory holding costs, faster throughput, reduced handling.

  • Disadvantages: Requires precise coordination, high IT investment, not suitable for all products.

• Outsourcing in SCM (3PL/4PL):

  • Importance: Allows companies to focus on core competencies, leverage external expertise, achieve cost savings.

  • Benefits: Cost reduction, improved service, access to global networks, flexibility.

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

• Expenditure vs. Opportunities:

  • Expenditure: Viewing SCM initiatives as costs to be minimized (traditional view).

  • Opportunities: Viewing SCM as a strategic investment that drives revenue, market share, and competitive advantage (modern view).

3.4 Evolution of Systems

• Development from MRP to ERP to Integrated SCM:

  • MRP (Material Requirements Planning): Focus on material planning for manufacturing (dependent demand). Uses BOM, inventory records, master production schedule.

  • MRP II (Manufacturing Resource Planning): Extends MRP to integrate all manufacturing resources (labor, machines, finance). Adds capacity planning, shop floor control.

  • ERP (Enterprise Resource Planning): Integrates all enterprise functions (finance, HR, sales, manufacturing) into a single system with centralized database. Scope beyond manufacturing.

  • Integrated SCM: Extends beyond enterprise to include external partners (suppliers, customers, logistics providers). Focus on collaboration, visibility, and synchronization across the entire chain.

• Link between SCM and E-Business:

  • E-Procurement: Online purchasing, auctions, catalogs.

  • E-Marketplaces: Platforms for spot buying/selling.

  • Visibility & Collaboration: Real-time data sharing via web portals, EDI, APIs.

  • Enables faster transactions, reduced transaction costs, better demand-supply matching.

[!TIP] Exam Focus: Bull-Whip effect (causes/mitigation) and MRP→ERP→SCM evolution are very high priority. Cross-docking disadvantages and outsourcing risks are high priority.

4.0 INVENTORY MANAGEMENT MODELS

4.1 Economic Order Quantity (EOQ) Model

Assumptions:

• Demand rate \(D\) is known, constant, and independent.

• Replenishment is instantaneous (order arrives all at once).

• No shortages allowed.

• Ordering cost \(S\) per order is fixed.

• Holding/carrying cost \(H\) per unit per year is fixed.

• Price per unit is constant (no discounts).

Derivation & Formula:

Total Annual Cost = Ordering Cost + Holding Cost

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

Minimize TC by differentiating w.r.t \(Q\) and setting to zero:

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

Key Calculations (Given \(D, S, H\)):

1. Optimum Order Quantity: \(Q^* = \sqrt{\frac{2DS}{H}}\)

2. Minimum Total Annual Cost: \(TC^* = \sqrt{2DSH}\)

3. Optimum Number of Orders per Year: \(N^* = \frac{D}{Q^*}\)

4. Optimum Time Between Orders (Cycle Time): \(T^* = \frac{365}{N^*}\) days or \(T^* = \frac{Q^*}{D}\) years.

[!TIP] Exam Focus: EOQ calculations are very high priority. Ensure units consistency (e.g., if \(H\) is per month, convert \(D\) to monthly). \(H\) often given as % of unit cost: \(H = i \times C\), where \(i\) = carrying rate, \(C\) = unit cost.

4.2 EOQ Model Variations

• Quantity Discounts:

  • All-Units Discount: Discount applies to entire order quantity if \(Q \ge\) breakpoint.

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

  • Decision Rule: Compute EOQ at each price level. If EOQ is feasible (within that price break's range), compute TC. Also compute TC at each breakpoint. Choose the price break with lowest TC.

• Shortages Allowed (Backordering):

  • Assumes backorders are allowed and filled later.

  • Optimal order quantity \(Q^*\) is larger than EOQ.

  • Maximum backorder level \(S^* = Q^* \sqrt{\frac{H}{H+P}}\), where \(P\) = shortage/penalty cost per unit per year.

  • Total cost includes shortage cost.

4.3 Inventory Classification & Analysis

• ABC Analysis (Pareto Principle):

  • Classify items based on annual consumption value (unit cost × annual usage).

  • A-class: ~15-20% items accounting for ~70-80% value. Tight control, frequent review.

  • B-class: ~30% items accounting for ~15-25% value. Normal control.

  • C-class: ~50-55% items accounting for ~5-10% value. Simple control, bulk ordering.

• VED Analysis (for spares/maintenance):

  • Based on criticality/importance of item to operations.

  • V (Vital): No operation without it. High stock.

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

  • D (Desirable): Can be managed without. Low stock.

• Advantages of ABC/VED:

  • Focused management attention on critical items.

  • Optimizes inventory investment.

  • Reduces stockouts for critical items.

  • Simplifies inventory control systems.

[!TIP] Exam Focus: ABC/VED explanation with advantages is high priority. Quantity discount decision rule is also high priority.

5.0 QUEUING THEORY (WAITING LINE ANALYSIS)

5.1 Basic Queuing Structure

Components:

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

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

3. Queue Discipline: Order of service (FIFO, LIFO, SIRO, Priority, Random).

4. System Capacity: Finite or infinite waiting room.

5. Customer Population: Finite or infinite source.

Queue Disciplines:

• 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 criteria (urgency, customer type).

• Random: Any customer selected randomly.

5.2 Poisson Arrival & Exponential Service Models (M/M/1)

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

• Parameters:

  • \(\lambda\) = average arrival rate (customers/unit time).

  • \(\mu\) = average service rate (customers/unit time).

  • Utilization Factor: \(\rho = \frac{\lambda}{\mu}\) (must be \(\rho < 1\) for steady state).

• Key Performance Measures (Steady-State):

  • \(P_n\) = Probability of exactly \(n\) customers in system.

  • \(L = \frac{\rho}{1-\rho}\) = Avg. number of customers in system (waiting + being served).

  • \(L_q = \frac{\rho^2}{1-\rho}\) = Avg. number of customers in queue.

  • \(W = \frac{1}{\mu - \lambda}\) = Avg. time in system (waiting + service).

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

  • \(P_0 = 1 - \rho\) = Probability system is empty.

Probability Calculations for Exponential Service:

• Service time \(T_s\) follows Exponential(\(\mu\)).

• Memoryless property: \(P(T_s > t + s | T_s > s) = P(T_s > t)\).

• Probability service time exceeds \(t\):

  \[ \boxed{P(T_s > t) = e^{-\mu t}} \]

• Probability service time is less than \(t\): \(P(T_s \le t) = 1 - e^{-\mu t}\).

5.3 Poisson Arrival Process

• Arrivals follow Poisson distribution with mean \(\lambda t\) over interval \(t\).

• Probability of exactly \(k\) arrivals in interval \(t\):

  \[ \boxed{P(k \text{ arrivals in } t) = \frac{e^{-\lambda t} (\lambda t)^k}{k!}} \]

• Inter-arrival times follow Exponential distribution with mean \(1/\lambda\).

[!TIP] Exam Focus: M/M/1 formulas (L, Lq, W, Wq) and exponential probability \(P(T>t) = e^{-\mu t}\) are very high priority. Poisson probability calculation is medium priority.

6.0 PROJECT MANAGEMENT (PERT/CPM)

6.1 Network Diagram Construction

• Activity-on-Arrow (AOA): Arrows represent activities, nodes represent events (milestones). Requires dummy activities to show dependencies.

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

• Network Logic:

  • Concurrent Activities: Can proceed simultaneously (no dependency).

  • Sequential Activities: Must follow one after another.

  • Dummy Activity: Zero duration, used in AOA to show dependency without consuming time/resources.

• Forward Pass: Calculate Earliest Start Time (EST) and Earliest Finish Time (EFT) for each activity.

  • EST of first activity = 0.

  • EFT = EST + duration.

  • EST of next activity = max(EFT of all immediate predecessors).

• Backward Pass: Calculate Latest Start Time (LST) and Latest Finish Time (LFT).

  • LFT of last activity = project completion time.

  • LST = LFT - duration.

  • LFT of previous activity = min(LST of all immediate successors).

6.2 Critical Path Method (CPM)

• Deterministic Time Estimates: Activity durations are known with certainty.

• Critical Path: Longest path through the network. Determines minimum project duration.

• Identification: Activities with Zero Total Float.

  • Total Float (TF): \(TF = LST - EST = LFT - EFT\). Slack time available without delaying project.

  • Free Float (FF): \(FF = EST_{next} - EFT\). Slacks without delaying successors.

• Significance of Critical Path:

  • Any delay in a critical activity directly delays project completion.

  • Management must focus resources on critical activities.

  • Non-critical activities have float; some delay is tolerable.

6.3 Program Evaluation and Review Technique (PERT)

• Probabilistic Time Estimates: Activity durations are uncertain, described by three estimates:

  • Optimistic Time (\(a\)): Minimum time if everything goes well.

  • Most Likely Time (\(m\)): Most realistic estimate.

  • Pessimistic Time (\(b\)): Maximum time if major problems occur.

• Expected Time (te):

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

• Variance (\(\sigma^2\)):

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

6.4 Project Completion Probability (PERT)

1. Compute expected project duration (\(T_E\)) as length of critical path (sum of \(t_e\)).

2. Compute project variance (\(\sigma_p^2\)) as sum of variances of activities on critical path.

3. For a due date \(T_D\), compute standard normal variable:

   \[ Z = \frac{T_D - T_E}{\sqrt{\sigma_p^2}} \]

4. Find probability \(P(Z \le z)\) using standard normal table.

   • Probability of completing by due date = \(P(Z \le z)\).

   • Probability of completing after due date = \(1 - P(Z \le z)\).

6.5 PERT vs. CPM

6.6 Phases of Project Management

1. Initiation: Define project scope, objectives, stakeholders.

2. Planning: Develop WBS, schedule (network), budget, resource plan, risk management plan.

3. Execution: Coordinate resources, carry out tasks, manage stakeholders.

4. Monitoring & Controlling: Track progress, manage changes, ensure alignment with plan (EVM, schedule control).

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

[!TIP] Exam Focus: PERT vs CPM comparison is very high priority. PERT probability calculation using Z-score is very high priority. Critical path significance and network diagram construction are high priority.

7.0 ADVANCED OR TOPICS & DECISION ANALYSIS

7.1 Game Theory

• Pure Strategy: Player chooses a single strategy (row/column) with certainty.

• Mixed Strategy: Player chooses strategies with probabilities (randomized choice).

• Basic Assumptions:

  1. Finite number of players (usually 2-person zero-sum).

  2. Players are rational and seek to maximize their payoff.

  3. All payoffs are known to all players.

  4. Decisions are made simultaneously or without knowledge of opponent's move.

  5. Game is played once (or iterated with same strategy).

• Saddle Point: Cell in payoff matrix where row minimum equals column maximum. Value of game \(V\). Pure strategy solution exists if maximin = minimax.

• Value of the Game: Expected payoff to the row player under optimal play.

7.2 Dominance Rules

• Row Dominance: Row \(i\) dominates row \(j\) if \(a_{ik} \ge a_{jk}\) for all \(k\) (for maximization) and \(>\) for at least one \(k\).

• Column Dominance: Column \(i\) dominates column \(j\) if \(a_{ki} \le a_{kj}\) for all \(k\) (for maximization) and \(<\) for at least one \(k\).

• Use: Dominated rows/columns can be deleted from payoff matrix, reducing problem size without losing optimal solution.

• Can be applied iteratively.

7.3 Heuristic & Meta-Heuristic Algorithms

• Heuristic: Rule-of-thumb, problem-specific, provides good (but not necessarily optimal) solution quickly. No guarantee of optimality.

  • Example: Nearest Neighbor for TSP.

• Meta-Heuristic: High-level, problem-independent framework that guides heuristics to explore solution space. Can escape local optima.

  • Genetic Algorithms (GA): Inspired by evolution (selection, crossover, mutation).

  • Simulated Annealing (SA): Inspired by annealing in metallurgy; accepts worse solutions with probability to escape local minima.

  • Tabu Search (TS): Uses memory (tabu list) to avoid cycling and explore new areas.

• Used for NP-hard problems where exact methods are computationally infeasible.

[!TIP] Exam Focus: Pure/Mixed strategies and assumptions are medium priority. Dominance rules are medium priority. Heuristic vs Meta-heuristic definition with examples is medium priority (often a short note).