Systems Engineering (ME-703 (C)) - Unit 4 Short Notes

UNIT 4: SYSTEMS ENGINEERING - EXAM-DRIVEN SHORT NOTES

I. LINEAR PROGRAMMING (LP)

Problem Formulation

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

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

$$\text{Maximize/Minimize } Z = c_1x_1 + c_2x_2 + ... + c_nx_n$$

• Constraints: Linear inequalities/equations representing resource limits, demand, etc.

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

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

[!TIP] Exam Focus: Translation of word problems (like skilled/semi-skilled labor constraints) into standard LP form is frequently tested.

Simplex Method (Maximization)

1. Convert to Standard Form: Add slack variables ($$\displaystyle S_i $$) for $\le$ constraints to turn them into equations. Surplus variables for $\ge$.

2. Initial Simplex Tableau: Set up matrix with coefficients of decision and slack variables. Basic variables = slack variables initially.

3. Optimality Test: Calculate $$\displaystyle C_j - Z_j $$ for all non-basic variables. If all $\le 0$, current solution is optimal.

4. Pivot Operation: Entering variable = most positive $$\displaystyle C_j - Z_j $$. Leaving variable = minimum positive ratio (RHS / pivot column coefficient). Perform row operations to make pivot element 1 and other elements in column 0.

5. Iterate until optimality condition met.

Interpretation of Final Tableau:

• Optimal Solution: Values in RHS column for basic decision variables.

• Optimal Objective Value (Z): Value in RHS of Z-row.

• Shadow Price (Dual Value): Value in Z-row under slack variable column. Indicates marginal value of one additional unit of that resource.

• Reduced Cost: $$\displaystyle C_j - Z_j $$ for non-basic variables. If >0, introducing that variable would increase Z.

Special Cases:

• Unbounded Solution: If all $$\displaystyle C_j - Z_j > 0 $$ but pivot column coefficients $\le 0$. Resource not limiting.

• Multiple Optimal Solutions: If a non-basic variable has $$\displaystyle C_j - Z_j = 0 $$ at optimum. Infinite solutions along an edge.

• Infeasibility: No feasible solution exists (e.g., contradictory constraints). Identified by artificial variables in basis with positive value at end (Big-M/Two-phase method).

II. TRANSPORTATION PROBLEMS

Initial Basic Feasible Solution (IBFS) Methods

Degeneracy

• Definition: An IBFS or solution during MODI is degenerate if number of occupied cells $$\displaystyle < (m + n - 1) $$. Means one or more basic variables are zero.

• Cause: Unusual allocation pattern during IBFS (e.g., NWC) or during iteration.

• Resolution: Introduce a very small $$\displaystyle \epsilon > 0 $$ in the zero cell to make it occupied for calculation purposes. This cell is treated as a basic variable with value $\epsilon$.

Transportation with Penalties/Unfulfilled Demand

• Add a dummy destination (if demand > supply) or dummy source (if supply > demand).

• Cost for dummy cells = penalty cost for unfulfilled requirement/supply.

• Solve as a balanced transportation problem. Allocation to dummy cell indicates amount of shortage/surplus.

Optimality Test (MODI / u-v Method)

1. For occupied cells: $$\displaystyle u_i + v_j = c_{ij} $$.

2. Set $$\displaystyle u_1 = 0 $$ (or any value), solve for all $$\displaystyle u_i, v_j $$.

3. For unoccupied cells, calculate $$\displaystyle C_{ij} - (u_i + v_j) $$.

4. Optimality: If all $$\displaystyle C_{ij} - (u_i + v_j) \ge 0 $$, solution is optimal.

5. Improvement: If any $$\displaystyle C_{ij} - (u_i + v_j) < 0 $$, select most negative cell as incoming. Form loop with occupied cells. Adjust allocations (+/- $\theta$) along loop. Determine $\theta$ (min value on -ve side). New solution.

III. INVENTORY MANAGEMENT

Economic Order Quantity (EOQ) Model

• Assumptions: Constant & known demand (D), instantaneous replenishment, no shortages, fixed ordering cost (S), constant holding cost per unit per year (H).

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

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

Differentiate w.r.t Q and set to zero.

• EOQ Formula:

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

• Key Metrics:

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

  • Cycle time (time between orders) = $$\displaystyle Q^* / D $$ (in years) or $$\displaystyle 365 \times (Q^*/D) $$ days.

  • Total Annual Cost at EOQ = $\sqrt{2DSH}$

[!TIP] Unit Consistency: Ensure D, S, H are in consistent time units (usually annual). Convert if demand is monthly/weekly.

EOQ with Quantity Discounts

• All-Units Discount: Entire order cost reduced to lower price if order quantity ≥ breakpoint.

• Incremental Discount: Only units above breakpoint get lower price.

• Decision Procedure:

  1. Calculate EOQ at lowest unit cost ($$\displaystyle c_{min} $$). If feasible (≥ breakpoint), it's optimal.

  2. If not feasible, calculate EOQ at each price break (using corresponding H = i% of unit cost). Feasible ones are candidates.

  3. Calculate Total Cost (including purchase cost) for all feasible EOQs and all breakpoints just below infeasible EOQs.

  4. Choose quantity with minimum total cost.

ABC Analysis

• Principle: Classify inventory items based on Annual Usage Value (AUV = Annual Demand × Unit Cost).

• Categories:

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

  • B Items: ~15-20% of value, ~20-30% of items. Moderate control.

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

• Purpose: Prioritize management efforts and capital investment.

VED Analysis (Spare Parts)

• Principle: Classify based on criticality to operations.

  • V (Vital): No substitute, stoppage if unavailable. Highest priority.

  • E (Essential): Important, but some short-term substitute/repair possible.

  • D (Desirable): Not critical, can be stocked minimally or procured quickly.

• Comparison with ABC: ABC is financial (value-based), VED is functional (criticality-based). Often used together (e.g., VED for critical spares, ABC within each VED class).

Carrying vs. Ordering Costs

IV. SUPPLY CHAIN MANAGEMENT (SCM)

Bull-Whip Effect

• Definition: Demand signal distortion as it moves upstream (from Retailer → Distributor → Manufacturer → Supplier). Small demand fluctuations at consumer end cause large order variability at supplier end.

• Causes:

  1. Demand Forecast Updating: Each echelon forecasts based on orders, not end-consumer demand.

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

  3. Price Fluctuations: Forward buying during promotions/discounts.

  4. Rationing & Gaming: Quota allocation based on orders, leading to false ordering.

• Consequences: Excessive inventory, poor capacity utilization, lost sales, inefficiency.

• Mitigation:

  • Information Sharing: Sharing point-of-sale (POS) data across chain.

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

  • Reducing Lead Times: Faster response, less need for safety stock.

  • Eliminating incentives for forward buying and order gaming.

Supply Chain Flows

Logistics in SCM

• Inbound Logistics: Activities from suppliers to production. Includes procurement, transportation, receiving, warehousing of raw materials. Goal: Ensure smooth, cost-effective material flow into the firm.

• Outbound Logistics: Activities from production to customer. Includes finished goods storage, order processing, transportation, delivery. Goal: Ensure timely, accurate, cost-effective product delivery to the customer. Directly impacts service level and customer satisfaction.

Cross-Docking

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

• Process: Inbound trucks → Dock → Sorting/Consolidation → Outbound trucks. Inventory "dwell time" is hours, not days.

• Requirements: Precise coordination, advanced IT systems (WMS, TMS), synchronized schedules, pre-tagged/palletized goods.

• Advantages: Reduced inventory holding costs, handling costs, and storage space; faster throughput.

• Disadvantages: High coordination complexity, requires high volume and reliability, less flexibility for errors.

Evolution: MRP → ERP → SCM

SCM and E-Business

• Linkages: E-business technologies enable SCM integration.

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

  • E-Logistics: Online tracking, carrier selection, freight payment.

  • Online Marketplaces: B2B exchanges, collaborative platforms.

• Impact: Reduces transaction costs, improves information flow speed/accuracy, enables new collaboration models (CPFR), but increases competition and requires integration.

Role of Inventory in SCM

• Primary Role: Buffer against uncertainty (demand variability, supply lead time variability).

• Decoupling Point: Location in chain where push (forecast-driven) meets pull (demand-driven). Inventory position defines this point.

• Trade-off: Higher inventory → Better service level, but higher holding cost. SCM aims to reduce total chain inventory through coordination, not just shift it.

Outsourcing in SCM

• Strategic Importance: Focus on core competencies, access to expertise/technology, cost reduction (labor, infrastructure), scalability, risk sharing.

• Risks: Loss of control, dependency on supplier, quality issues, knowledge drain, hidden costs, security risks. Requires strong partnership management (SLAs, relationship management).

Expenditure & Opportunities in SCM

• Major Cost Areas: Transportation (largest), Inventory carrying, Facilities (warehouses), Information systems, Administration.

• Opportunities:

  • Technology: IoT, AI/ML for forecasting, blockchain for traceability, advanced analytics.

  • Collaboration: VMI, CPFR, strategic partnerships.

  • Sustainability: Green logistics, reverse logistics, circular economy.

  • Network Design: Optimizing facility locations, mode selection.

V. PROJECT MANAGEMENT (PERT/CPM)

PERT vs. CPM

Network Diagrams

• Activity-on-Node (AON / Precedence Diagramming): Activities as nodes, arrows show dependencies. Most common. Logical relationships: FS (Finish-Start), SS (Start-Start), FF (Finish-Finish), SF (Start-Finish).

• Activity-on-Arrow (AOA): Activities as arrows, nodes as events. Requires dummy activities (dashed arrows, zero time/cost) to maintain logic and uniqueness of node numbering.

• Drawing from Activity List: Identify immediate predecessors. Draw nodes/arrows respecting dependencies. Ensure no loops.

Critical Path Method (CPM)

1. Forward Pass: Calculate Earliest Start Time (EST) and Earliest Finish Time (EFT).

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

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

   • Project duration = max(EFT of terminal activities).

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

   • $$\displaystyle LFT_j = \min(LST_j) $$ for all immediate successors $j$ of $i$.

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

   • For terminal nodes, $$\displaystyle LFT = EFT $$ (project duration).

3. Float/Slack:

   • Total Float (TF): $$\displaystyle TF_i = LST_i - EST_i = LFT_i - EFT_i $$. Time an activity can be delayed without delaying project.

   • Free Float (FF): $$\displaystyle FF_i = EST_j(\text{successor}) - EFT_i $$. Delay without delaying early start of successor.

   • Independent Float: Unused time within float.

4. Critical Path: Path with zero total float. Longest path through network. Determines minimum project duration. Activities on CP are critical; any delay delays project.

PERT Time Estimates & Variance

• Optimistic (O): Time if everything goes better than expected.

• Pessimistic (P): Time if everything goes worse than expected.

• Most Likely (M): Most realistic estimate.

• Expected Activity Time:

$$TE = \frac{O + 4M + P}{6}$$

• Activity Variance:

$$\sigma_a^2 = \left(\frac{P - O}{6}\right)^2$$

• Project Variance ($$\displaystyle \sigma_p^2 $$): Sum of variances of activities on the critical path (assuming independence).

Project Duration Probability

• Assume project completion time follows Normal Distribution with mean = $$\displaystyle TE_{project} $$ (sum of TE on CP) and variance = $$\displaystyle \sigma_p^2 $$ (sum of $$\displaystyle \sigma_a^2 $$ on CP).

• Z-score:

$$Z = \frac{D - TE_{project}}{\sqrt{\sigma_p^2}}$$

where $D$ = due date/target completion time.

• Probability of completion by D: $P(Z \le \text{calculated Z})$ from standard normal table.

• Probability of exceeding D: $1 - P(Z \le \text{Z})$.

Heuristic & Meta-Heuristic Algorithms

• Heuristics: Rule-of-thumb, problem-specific, quick, "good enough" solutions. Not guaranteed optimal.

  Examples: Nearest Neighbor (TSP), First-Come-First-Served (scheduling), Lowest Cost Rule (transportation).

• Meta-Heuristics: Higher-level, general-purpose frameworks that guide heuristics to escape local optima. Can handle large, complex problems.

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

• Application in Projects: Resource leveling/allocation, project scheduling with multiple constraints, portfolio optimization.

Network Logics

• Precedence Relationships: Definition of which activities must precede others (FS, SS, FF, SF).

• Dummy Activity: Used in AOA to show dependency without consuming time/resource. Maintains network consistency (unique node numbering, correct logic).

• Complex Constraints: Lag/lead times (e.g., "Start Activity B 5 days after Start of A" = SS+5). Often handled in AON software.

• Avoiding Loops: Network must be acyclic (no circular dependencies).

VI. QUEUEING THEORY

Basic Queueing Model (Kendall Notation: A/B/c)

• A: Arrival process (e.g., M = Markov/Poisson, D = Deterministic, G = General).

• B: Service time distribution (M, D, G).

• c: Number of parallel servers.

• Additional Parameters: Queue capacity (K), population size (N). Default: $\infty$.

• Common Model: M/M/1 (Poisson arrivals, Exponential service, 1 server).

Poisson Arrivals & Exponential Service

• Poisson Process (Arrivals): Probability of $k$ arrivals in interval $t$:

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

where $\lambda$ = mean arrival rate (per unit time).

• Exponential Distribution (Service): Probability service time $$\displaystyle > t $$:

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

where $\mu$ = mean service rate (per unit time). **Memoryless property.**

• Utilization Factor:

$$\rho = \frac{\lambda}{\mu}$$

Must be $$\displaystyle \rho < 1 $$ for steady state (M/M/1).

Probability Calculations (M/M/1)

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

• Probability service time > t: $$\displaystyle P(T > t) = e^{-\mu t} $$ (directly from exponential distribution).

• Probability of exactly k arrivals in time t: Use Poisson formula with $\lambda t$.

Queue Disciplines

VII. GAME THEORY

Pure Strategies

• Definition: A player chooses a single specific action with certainty.

• Payoff Matrix: Rows = Player A strategies, Columns = Player B strategies. Entries = payoff to row player (A) (often zero-sum).

• Saddle Point: Cell where row minimum = column maximum = game value.

  • Maximin (A): Maximize own minimum payoff. $$\displaystyle \max_i \min_j a_{ij} $$

  • Minimax (B): Minimize own maximum loss (or maximize A's minimum gain). $$\displaystyle \min_j \max_i a_{ij} $$

  • If Maximin = Minimax, saddle point exists. Strategies are pure optimal.

• Dominant Strategy: Strategy that yields a higher payoff regardless of opponent's choice. If one exists for a player, it is optimal.

Mixed Strategies

• Definition: Player chooses a pure strategy according to a probability distribution.

• Expected Payoff: Sum over all strategy pairs: (probability A chooses i) × (probability B chooses j) × (payoff $$\displaystyle a_{ij} $$).

• Solving 2x2 Games (No Saddle Point):

  • Algebraic: Let A play strategy 1 with prob p, 2 with (1-p). Set B's expected payoff equal for both of B's pure strategies to make B indifferent. Solve for p. Compute game value.

  • Graphical: Plot A's expected payoff vs p for each of B's pure strategies. Find intersection point (minimax point for A).

• Value of Game: Expected payoff to row player (A) when both use optimal mixed strategies.

Basic Assumptions

1. Rational Players: Each player aims to maximize their own payoff.

2. Fixed Payoff Matrix: Payoffs are known, constant.

3. Simultaneous Move or Sequential with Known Actions: Players choose without knowledge of opponent's current choice (or move in sequence with full information).

4. Common Knowledge: Rules, strategies, and payoffs are known to all players.

5. Zero-Sum (often in basic OR): One player's gain is exactly the other's loss. Sum of payoffs in each cell = 0.

Dominance Rule (Reducing Game Size)

• Strict Domination: Strategy $i$ dominates strategy $j$ for a player if payoff($i$, any opponent strategy) > payoff($j$, same opponent strategy). $j$ can be eliminated.

• Weak Domination: payoff($i$, any) $\ge$ payoff($j$, any), and > for at least one. $j$ can be eliminated if the dominated strategy is never a best response.

• Iterative Elimination: Repeatedly remove dominated rows/columns to simplify matrix before solving.

VIII. ADDITIONAL TOPICS (Lower Frequency)

Dominance Rule in Transportation

• Similar to game theory. A row (source) is dominated if for every column, its cost is $\ge$ cost of another row. Can be eliminated.

• A column (destination) is dominated if for every row, its cost is $\ge$ cost of another column.

• Modified Rule (Opportunity Cost): Sometimes, $$\displaystyle c_{ij} \ge c_{ik} + c_{kj} $$ for all k indicates dominance (indirect route cheaper). Used less frequently in exams.

Short Notes: Heuristic & Meta-Heuristic Algorithms

• Heuristics: Problem-specific, fast, intuitive rules providing feasible solutions quickly. No optimality guarantee. Example: VAM is a heuristic for transportation.

• Meta-Heuristics: General, high-level frameworks that guide search processes (often using heuristics) to explore solution space globally and avoid local optima. Stochastic, iterative. Examples: Genetic Algorithms (mutation, crossover), Simulated Annealing (probabilistic acceptance of worse solutions), Tabu Search (memory of recent moves).

• Use: NP-hard problems (TSP, scheduling, network design) where exact methods are too slow.

Short Notes: Network Logics

• Precedence: The fundamental constraint defining activity sequence (e.g., "Foundation" must finish before "Walls" start - FS relationship).

• Dummy Activity: In AOA, a zero-duration activity used solely to show dependency (e.g., two activities share a common predecessor but have no direct relationship).

• Complex Dependencies: Lag/lead times (e.g., "Start B 5 days after Start of A" = SS+5). Modern software (AON) handles this directly.

• Key Principle: Network must be acyclic (no loops) and connected (all activities linked).