UNIT 3: OPERATION RESEARCH & SUPPLY CHAIN - EXAM-FOCUSED NOTES

1. LINEAR PROGRAMMING (LP) & SIMPLEX METHOD

Formulation of LP Problems

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

• Objective Function: Linear function to be maximized (profit) or minimized (cost).

$$Z = c_1x_1 + c_2x_2 + ... + c_nx_n$$

• Constraints: Linear inequalities/equations representing resource limits.

$$a_{11}x_1 + a_{12}x_2 + ... \le, =, \ge b_1$$

• Non-negativity: $$\displaystyle x_1, x_2, ... \ge 0 $$.

• Standard Form (Maximization): All constraints as ≤, RHS ≥ 0, objective as Max Z.

[!TIP] Exam Pattern: Direct formulation from word problems is a 14-mark question. Identify variables, write objective, convert story to constraints, add $$\displaystyle x_i \ge 0 $$.

Graphical Method (2 Variables)

1. Plot each constraint as a line.

2. Identify feasible region (intersection satisfying all constraints).

3. Plot iso-profit line for objective function.

4. Move line parallelly to find optimal corner point (extreme point).

5. Optimal solution always at a corner of feasible region.

Simplex Method (Maximization, ≤ Constraints)

Step-by-Step Algorithm:

1. Convert to Standard Form: Add slack variables ($$\displaystyle s_i \ge 0 $$) to ≤ constraints.

   Example: $$\displaystyle 3x_1 + 2x_2 \le 6 $$ becomes $$\displaystyle 3x_1 + 2x_2 + s_1 = 6 $$.

2. Initial Basic Feasible Solution (IBFS): Set non-basic variables ($$\displaystyle x_j $$) to 0. Solve for basic variables ($$\displaystyle s_i $$).

3. Set up Simplex Tableau: Rows for constraints, columns for all variables. Bottom row for net evaluation row ($$\displaystyle C_j - Z_j $$).

4. Optimality Test: If all $$\displaystyle C_j - Z_j \le 0 $$ (for Max), current solution is optimal.

5. Pivot Operation (if not optimal):

   • Entering Variable: Most positive $$\displaystyle C_j - Z_j $$.

   • Leaving Variable: Minimum positive ratio (RHS / pivot column coefficient). Tie-breaking: Choose arbitrarily or use smallest subscript rule.

   • Pivot Element: Intersection of entering column & leaving row.

   • Row Operations: Convert pivot element to 1, other elements in column to 0.

6. Repeat until optimal.

7. Interpret Final Tableau:

   • Optimal values: RHS of basic variable rows.

   • Optimal Z: Bottom RHS cell.

   • Shadow Price/Dual Value: Value in bottom row under slack variable column (marginal value of RHS resource).

[!TIP] Common Pitfalls: Forgetting to calculate $$\displaystyle Z_j $$ row correctly, incorrect ratio test (only positive ratios), missing non-negativity.

Handling Artificial Variables (≥ or = Constraints)

• Big-M Method: Add artificial variable ($$\displaystyle a_i $$) with very high penalty cost (-M for Max, +M for Min) in objective.

• Two-Phase Method:

  • Phase I: Minimize sum of artificial variables. If min > 0 → infeasible.

  • Phase II: Use feasible basis from Phase I (remove artificial vars/columns), solve original objective.

Sensitivity Analysis (Post-Optimality)

• Shadow Price: Valid within allowable range for RHS (found in final tableau under RHS column limits).

• Objective Coefficient Range: Range where current basis remains optimal. Change outside range → re-solve.

• Key Insight: Shadow price = change in optimal Z per unit increase in RHS, only if within allowable range.

2. TRANSPORTATION PROBLEMS

Problem Structure

• Origins (m): Sources with Supply ($$\displaystyle S_i $$).

• Destinations (n): Sinks with Demand ($$\displaystyle D_j $$).

• Balanced: $$\displaystyle \sum S_i = \sum D_j $$. Unbalanced: Add Dummy Row/Column with zero cost.

• Objective: Minimize total transportation cost: $$\displaystyle \sum \sum c_{ij} x_{ij} $$.

Finding Initial Basic Feasible Solution (IBFS)

Optimality Test: MODI (UV) Method

Steps:

1. For IBFS with m+n-1 allocations, assign u_i (row potentials) & v_j (col potentials).

   • Set $$\displaystyle u_1 = 0 $$ (or any).

   • For each basic cell $(i,j)$: $$\displaystyle u_i + v_j = c_{ij} $$.

2. Compute opportunity cost for non-basic cells: $$\displaystyle \Delta_{ij} = c_{ij} - (u_i + v_j) $$.

3. Optimality Test: If all $$\displaystyle \Delta_{ij} \ge 0 $$ → optimal.

4. If not optimal: Select most negative $$\displaystyle \Delta_{ij} $$ as entering cell.

5. Loop Formation: Form closed loop from entering cell, alternating +/-.

6. Allocation Adjustment: Find minimum allocation in - cells ($\theta$). Adjust loop: +θ to + cells, -θ to - cells.

7. New solution → repeat from Step 1.

[!TIP] Degeneracy: If allocations < m+n-1, introduce dummy allocation ($\epsilon$) in a zero-cost cell to complete loop.

Advanced Transportation Problems

• Transshipment: Intermediate nodes allowed. Solve by converting to standard TP (add dummy costs).

• Penalties for Unfulfilled Demand: Add dummy destination with cost = transport cost + penalty. Allocate to dummy if cheaper than fulfilling.

• Maximization Problem: Convert to minimization by subtracting all costs from a large constant (e.g., $$\displaystyle C_{max} $$).

3. SUPPLY CHAIN MANAGEMENT (SCM) CORE CONCEPTS

Definition & Objectives

• SCM: Network of organizations working together to produce & deliver value to end-customer.

• Objectives: Reduce costs, improve service, increase responsiveness, create competitive advantage.

Key Flows in SCM

Bull-Whip Effect

• Definition: Demand distortion as orders move upstream (retailer → distributor → manufacturer). Small demand change at consumer causes large order variance at supplier.

• Causes:

  1. Demand Forecast Updating: Each tier forecasts based on orders, not consumer demand.

  2. Order Batching: Large, infrequent orders to reduce ordering costs.

  3. Price Fluctuations: Forward buying during promotions.

  4. Rationing & Shortage Gaming: Customers over-order when supply is limited.

• Consequences: Excess inventory, poor customer service, inefficient production, high costs.

• Mitigation Strategies:

  • VMI (Vendor Managed Inventory): Supplier manages inventory at customer site.

  • CPFR (Collaborative Planning, Forecasting, Replenishment): Shared data & joint forecasts.

  • Stabilize Prices: Eliminate forward buying.

  • Reduce Lead Times: Faster response, less need to forecast.

  • Information Sharing: Point-of-Sale (POS) data to all tiers.

Logistics in SCM

[!TIP] Exam Focus: "Explain importance of inbound/outbound logistics" is a 7-mark question. Link to cost, service level, and competitive advantage.

Cross-Docking

• Definition: Products from inbound trucks are directly sorted & transferred to outbound trucks with minimal/no storage.

• Advantages: Reduced inventory & handling costs, faster throughput, lower warehousing space.

• Disadvantages: Requires precise coordination, high IT dependency, suitable only for high-volume, predictable goods.

Outsourcing in SCM

• Rationale: Focus on core competencies, reduce costs, gain expertise, improve flexibility.

• Benefits: Cost reduction, access to technology, risk sharing.

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

Expenditure vs. Opportunities

• Expenditure Focus: Viewing SCM as a cost center (minimize transportation, warehousing costs).

• Opportunity Focus: Viewing SCM as a strategic weapon to increase sales, market share, customer satisfaction (e.g., faster delivery = premium price).

4. INVENTORY MANAGEMENT

Role of Inventory

• Decouple processes (production vs. sales).

• Buffer against demand & supply uncertainty.

• Enable economies of scale (larger orders).

• Trade-off: Holding Cost vs. Ordering Cost vs. Shortage Cost vs. Service Level.

Economic Order Quantity (EOQ) Model

Assumptions:

1. Constant, known demand rate ($D$).

2. Instantaneous replenishment (order arrives all at once).

3. Fixed ordering cost ($S$) per order.

4. Fixed holding cost ($H$) per unit per year.

5. No shortages allowed.

Derivation & Formula:

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

Where:

• $D$ = Annual demand (units/year)

• $S$ = Ordering cost (Rs/order)

• $H$ = Holding cost (Rs/unit/year)

Key Calculations:

• Number of orders/year: $$\displaystyle N = D / Q^* $$

• Cycle time (time between orders): $$\displaystyle T = 1/N = Q^*/D $$ (years)

• Minimum Total Annual Cost (TAC):

$$TAC = \text{Purchase Cost} + \frac{D}{Q^*}S + \frac{Q^*}{2}H$$

(Purchase cost constant for fixed D, so minimize ordering + holding).

[!TIP] Exam Pattern: EOQ numericals are guaranteed. Watch units: if H is given as % of cost, compute $$\displaystyle H = i \times C $$ (i = carrying rate, C = unit cost). Convert monthly demand to annual if needed.

EOQ Variants

• Finite Production Rate (EPQ): Production rate $$\displaystyle P > D $$. Inventory builds gradually.

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

• Quantity Discounts:

  1. Compute EOQ for each price break using $$\displaystyle H = i \times \text{price} $$.

  2. Feasibility Check: If EOQ for a price break is outside its quantity range, use minimum quantity for that price.

  3. Compute TAC for all feasible quantities (EOQs & breakpoints).

  4. Select quantity with minimum TAC.

• Planned Shortages: When shortage cost ($p$) is finite.

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

Max inventory = $$\displaystyle Q^* \cdot \frac{p}{H+p} $$.

ABC Analysis (Pareto Principle)

• Classification by Annual Usage Value (AUV): $$\displaystyle AUV = \text{Annual Demand} \times \text{Unit Cost} $$.

• Categories:

  • A-items: ~70-80% of total AUV, ~10-20% of items. Tight control, frequent review.

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

  • C-items: ~5% of AUV, ~50-70% of items. Loose control, bulk ordering.

• Purpose: Prioritize management effort & resources.

VED Analysis (For Spare Parts)

• V (Vital): Stock-out stops production. High stock.

• E (Essential): Stock-out seriously affects production. Moderate stock.

• D (Desirable): Stock-out causes minor inconvenience. Low stock.

• Difference from ABC: ABC is monetary (value), VED is criticality (functional importance). Often used together (e.g., A-V items are most critical).

[!TIP] Advantages of ABC/VED: Efficient resource allocation, focused control, reduced inventory costs, better availability for critical items.

5. QUEUEING THEORY

Basic Concepts

• Arrival Process: Described by arrival rate $\lambda$ (avg. arrivals/unit time).

• Service Process: Described by service rate $\mu$ (avg. services/unit time).

• Queue Discipline: Rule for selecting next customer (FCFS most common).

• System Capacity: Max number of customers allowed (finite/infinite).

• Number of Servers (c): Single (c=1) or multiple.

Probability Distributions

• Poisson Arrivals: Probability of $k$ arrivals in time $t$:

$$P(k) = \frac{e^{-\lambda t} (\lambda t)^k}{k!}$$

*Mean number in time $$\displaystyle t = \lambda t $$.*

• Exponential Service: Probability service time > $t$:

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

*Mean service time = $1/\mu$.*

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

M/M/1 Queue (Single Server)

• Notation: M/M/1 = Poisson arrivals, Exponential service, 1 server, infinite capacity, FCFS.

• Utilization Factor: $$\displaystyle \rho = \lambda / \mu $$. Must have $$\displaystyle \rho < 1 $$ for steady state.

• Performance Measures:

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

  • $L$ = Avg. number in system = $$\displaystyle \frac{\rho}{1-\rho} $$

  • $$\displaystyle L_q $$ = Avg. number in queue = $$\displaystyle \frac{\rho^2}{1-\rho} $$

  • $W$ = Avg. time in system = $$\displaystyle \frac{1}{\mu - \lambda} $$

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

  • $$\displaystyle L = \lambda W $$, $$\displaystyle L_q = \lambda W_q $$ (Little's Law).

[!TIP] Exam Problems: Given $\lambda$ and $\mu$ (or avg. service time), compute probabilities or averages. Check $$\displaystyle \rho < 1 $$ first. For "more than t time" questions, use exponential service distribution directly.

6. PROJECT MANAGEMENT (PERT/CPM)

Network Diagram Construction

• Activity-on-Node (AON/PERT): Most common. Node = activity, arrow = precedence.

• Activity-on-Arrow (AOA): Arrow = activity, node = event (milestone).

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

• Logical Relationships:

  • FS (Finish-to-Start): Most common. B starts after A finishes.

  • SS (Start-to-Start), FF (Finish-to-Finish), SF (Start-to-Finish).

[!TIP] Network Logics: Ensure all predecessors are correctly shown before an activity. No dangling activities. Check for loops.

Critical Path Method (CPM)

• Deterministic times (known with certainty).

• Forward Pass (EST, EFT):

  • $$\displaystyle EST_i = \max(EFT_j) $$ for all immediate predecessors $j$.

  • $$\displaystyle EFT_i = EST_i + t_i $$.

  • Start node: $$\displaystyle EST=0 $$.

• Backward Pass (LST, LFT):

  • $$\displaystyle LFT_i = \min(LST_j) $$ for all immediate successors $j$.

  • $$\displaystyle LST_i = LFT_i - t_i $$.

  • End node: $$\displaystyle LFT = \text{project duration} $$.

• Float/Slack:

  • Total Float (TF): $$\displaystyle TF_i = LST_i - EST_i = LFT_i - EFT_i $$. Critical path = TF = 0.

  • Free Float (FF): $$\displaystyle FF_i = \min(EST_j) - EFT_i $$ (for successors $j$). Doesn't affect successors.

• Critical Path: Longest path through network. Delays here delay project.

Program Evaluation and Review Technique (PERT)

• Probabilistic times (uncertainty).

• Time Estimates per Activity:

  • $a$ = Optimistic (best case)

  • $m$ = Most Likely

  • $b$ = Pessimistic (worst case)

• Expected Time: $$\displaystyle \boxed{t_e = \frac{a + 4m + b}{6}} $$

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

• Project Duration ($$\displaystyle T_e $$): Sum of $$\displaystyle t_e $$ along critical path.

• Project Variance ($V$): Sum of $$\displaystyle \sigma^2 $$ along critical path (assumes independence).

• Probability of Completion by Due Date ($D$):

  1. Compute Standard Deviations: $$\displaystyle \sigma_p = \sqrt{V} $$.

  2. Compute Z-score: $$\displaystyle \boxed{Z = \frac{D - T_e}{\sigma_p}} $$

  3. Find probability from standard normal table for $Z$.

[!TIP] CPM vs PERT:

| Feature | CPM | PERT |

| :--- | :--- | :--- |

| Time | Deterministic | Probabilistic (a, m, b) |

| Focus | Time-Cost Trade-off | Time Uncertainty |

| Use | Construction, repetitive projects | R&D, new product development |

| Analysis | Crashing | Probability of completion |

7. OTHER IMPORTANT TOPICS & SHORT NOTES

Heuristic & Meta-Heuristic Algorithms

• Heuristic: Problem-specific rule-of-thumb. Fast, good (not optimal) solution. E.g., Nearest Neighbor for TSP.

• Meta-Heuristic: Problem-independent framework guiding heuristics. Explores solution space.

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

  • Simulated Annealing: Mimics cooling process, accepts worse moves to escape local optima.

  • Tabu Search: Uses memory (tabu list) to avoid cycling.

  • Ant Colony Optimization: Swarm intelligence, pheromone trails.

• Use When: Problems are NP-hard, large-scale, exact methods (like simplex) too slow.

Development: MRP → MRP II → ERP → SCM

Game Theory (Basics)

• Players: Decision-makers.

• Strategies: Possible actions.

• Payoff Matrix: Outcomes for each strategy combination.

• Pure Strategy: Single, definite choice.

• Mixed Strategy: Probability distribution over pure strategies.

• Dominance Rule: If strategy A yields better payoff than B for all opponent strategies, eliminate B.

• Assumptions: Rational players, known payoffs, simultaneous or sequential with perfect info.

Inventory in Logistics Supply Chain System

• Inventory is the physical link between different echelons (supplier → manufacturer → distributor → retailer).

• Bull-Whip Effect causes inventory amplification upstream.

• SCM Goal: Reduce total system inventory while improving customer service through coordination (VMI, CPFR), not just optimizing each echelon locally.

Final Exam Strategy:

1. LP/Simplex: Master tableau setup, pivot rules, interpretation.

2. Transportation: VAM for IBFS, MODI for optimality, degeneracy handling, penalty problems.

3. SCM: Bull-Whip (causes/mitigation), logistics types, cross-docking.

4. Inventory: EOQ formula & variants, ABC/VED.

5. Queueing: M/M/1 formulas, exponential/Poisson calculations.

6. PERT/CPM: Network diagram, forward/backward pass, PERT time estimates, Z-score probability.

7. Short Notes: Be precise: MRP→ERP→SCM, heuristics vs meta-heuristics, pure/mixed strategies.

All formulas boxed. All key terms bolded. All exam traps highlighted.