Systems Engineering (ME-703 (C)) - Unit 2 Short Notes

UNIT 2: OPERATIONS RESEARCH AND SUPPLY CHAIN MANAGEMENT IN SYSTEMS ENGINEERING

I. LINEAR PROGRAMMING (LP)

A. Problem Formulation and Modeling

• 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 (labor, material, time).

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

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

• Example (Product Mix): Maximize profit subject to labor and machine time constraints.

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EXAM FOCUS: Converting word problems into standard LP form is a high-frequency question. Clearly define variables, write objective, and identify all constraints with units.

B. Simplex Method

• Standard Form Conversion:

  • Maximization problem with $\le$ constraints.

  • Add slack variables ($$\displaystyle s_i \ge 0 $$) to convert inequalities to equalities.

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

• Iterative Procedure:

  1. Initial Basic Feasible Solution (IBFS): Set decision variables to 0; slack variables = RHS.

  2. Entering Variable: Choose non-basic variable with most positive coefficient in objective row (for maximization) in the simplex tableau.

  3. Leaving Variable: Compute Minimum Ratio Test (RHS / pivot column element, only for positive pivot elements). Smallest ratio determines leaving variable.

  4. Pivot Operation: Use elementary row operations to make pivot element = 1 and all other elements in pivot column = 0.

  5. Repeat until no positive coefficients remain in objective row (for maximization).

• Interpretation of Final Tableau:

  • Basic variables = solution values in RHS column.

  • Non-basic variables = 0.

  • Optimal objective value = RHS of objective row.

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COMMON PITFALL: Forgetting to convert to standard form (adding slack/surplus). In maximization, stop when all objective row coefficients are ≤ 0. In minimization, stop when all are ≥ 0.

C. Special Cases

• Infeasibility: No solution satisfies all constraints. Identified by artificial variables in final basis with positive value (using Big-M or Two-Phase method).

• Unboundedness: Objective can increase indefinitely. Occurs if entering variable's column has all non-positive elements (no leaving variable).

• Multiple Optima: Non-basic variable has zero coefficient in final objective row. Infinite solutions along the edge between two optimal extreme points.

II. TRANSPORTATION PROBLEMS

A. Initial Basic Feasible Solutions (IBFS)

• North-West Corner (NWC) Rule:

  1. Start at top-left cell (row 1, col 1).

  2. Allocate as much as possible: min(available supply, required demand).

  3. Adjust supply/demand, move right if demand exhausted, down if supply exhausted.

  4. Repeat until all allocations made.

• Vogel's Approximation Method (VAM):

  1. For each row/column, compute penalty = difference between two smallest costs.

  2. Select row/column with highest penalty.

  3. Allocate to cell with lowest cost in that row/column (min(supply, demand)).

  4. Adjust, cross out exhausted row/column, recalculate penalties.

  5. Repeat.

  • Aim: Minimize total transportation cost. Generally better than NWC.

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EXAM TIP: VAM steps are frequently asked. Always show penalty calculation table. NWC is simpler but may yield higher initial cost.

B. Degeneracy in Transportation Problems

• Definition: Number of allocated cells < $(m + n - 1)$. A basic feasible solution must have exactly $(m+n-1)$ independent allocations.

• Causes: Simultaneous satisfaction of a row supply and column demand during allocation.

• Resolution:

  1. Identify degenerate cell (allocation = 0 in a position that should be basic).

  2. Assign a very small epsilon (ε) to that cell (conceptually).

  3. Proceed with MODI/Stepping Stone method normally, treating ε as a positive allocation.

C. Transportation with Penalties for Unfulfilled Demand

• Formulation:

  • Add a dummy destination (or source) with zero transportation cost from all sources.

  • Set penalty cost for unfulfilled demand at dummy destination.

  • Total supply must equal total requirement (including dummy).

• Solution Approach: Solve as standard transportation problem. Allocation to dummy destination represents unfulfilled demand at given penalty.

III. INVENTORY MANAGEMENT MODELS

A. Economic Order Quantity (EOQ) Model

• Assumptions:

  • Constant, known demand rate ($D$ units/year).

  • Instantaneous replenishment (order arrives all at once).

  • No shortages allowed.

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

  • Constant holding cost ($H$ per unit per year).

• Key Formulas:

  • Optimal Order Quantity:

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

*   Total Annual Cost (TAC):  

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

    (where $C$ = unit purchase cost)

*   Number of Orders per Year: $$\displaystyle N = D / Q^* $$

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

• Variations:

  • Quantity Discounts: Compare TAC at EOQ and at each discount breakpoint. Choose quantity with lowest TAC.

  • Finite Production Rate (EPQ): When production rate ($P$) > demand rate ($D$).

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

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EXAM CRITICAL: Ensure holding cost (H) is annual. If given as % of unit cost, $$\displaystyle H = i \times C $$ where $i$ = carrying cost rate. Convert all time units to years.

B. Inventory Classification Systems

• ABC Analysis (Pareto Principle):

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

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

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

  • C-items: Remaining ~5% of value, ~50-60% of items. Loose control.

• VED Analysis (Criticality):

  • Vital (V): Items whose stoppage halts production. Highest priority.

  • Essential (E): Items whose shortage seriously affects efficiency.

  • Desirable (D): Items whose shortage is inconvenient but not critical.

• Comparative Advantages:

  • ABC: Focuses on cost reduction and inventory investment control.

  • VED: Focuses on availability and service level for critical spares/items.

  • Often used together: ABC for cost, VED for criticality.

IV. QUEUING THEORY

A. Fundamental Concepts (Kendall's Notation: A/B/c)

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

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

• c: Number of servers.

• Other parameters: System capacity, population size, service discipline (FIFO, LIFO, etc.).

B. M/M/1 Model (Single Server)

• Parameters:

  • Arrival rate: $\lambda$ (customers/unit time)

  • Service rate: $\mu$ (customers/unit time)

  • Utilization factor: $$\displaystyle \rho = \lambda / \mu $$ (must be $$\displaystyle \rho < 1 $$ for steady state).

• Performance Metrics:

  • $L$ = Avg. number of customers 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 = $$\displaystyle \frac{\rho}{\mu - \lambda} = \frac{L_q}{\lambda} $$

C. Probability Calculations

• Exponential Service Time:

  $$\displaystyle P(\text{service time} > t) = e^{-\mu t} $$

• Poisson Arrivals:

  $$\displaystyle P(N = n \text{ arrivals in time } t) = \frac{(\lambda t)^n e^{-\lambda t}}{n!} $$

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MEMORIZE: For M/M/1, $$\displaystyle W = \frac{1}{\mu - \lambda} $$ and $$\displaystyle W_q = \frac{\rho}{\mu - \lambda} $$. Often asked: "Probability service time > t" uses exponential distribution directly.

D. Queue Disciplines

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

• LIFO/LCFS: Last-In-First-Out.

• SIRO: Service In Random Order.

• Priority: Based on assigned priority (preemptive/non-preemptive).

• Impact: Discipline affects $$\displaystyle L_q $$ and $$\displaystyle W_q $$ but not $L$ and $W$ for M/M/1 (due to memoryless property).

V. PROJECT MANAGEMENT TECHNIQUES

A. Network Diagram Construction

• Activity-on-Arrow (AOA): Arrows represent activities, nodes are events. Requires dummy activities to maintain logic.

• Activity-on-Node (AON/PERT): Nodes represent activities, arrows show precedence. More common.

• Precedence Relationships:

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

  • SS (Start-to-Start): B starts after A starts.

  • FF (Finish-to-Finish): B finishes after A finishes.

  • SF (Start-to-Finish): B finishes after A starts (rare).

B. CPM vs PERT

• PERT Time Estimates:

  • Optimistic ($a$), Pessimistic ($b$), Most Likely ($m$).

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

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

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

• Float (Slack):

  • Total Float (TF): $$\displaystyle LS - ES = LF - EF $$. Time an activity can be delayed without delaying project.

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

  • Independent Float: Float considering only predecessors and successors.

C. Project Completion Probability

1. Find critical path and sum variances along it: $$\displaystyle \sigma_{CP}^2 = \sum \sigma_i^2 $$.

2. Assume project duration follows Normal distribution with mean $$\displaystyle T_{E(CP)} $$ and std. dev. $$\displaystyle \sigma_{CP} $$.

3. For due date $$\displaystyle T_d $$, compute Z-score:

$$Z = \frac{T_d - T_{E(CP)}}{\sigma_{CP}}$$

4. Find probability from standard normal table: $$\displaystyle P(T \le T_d) = \Phi(Z) $$.

D. Phases of Project Management

1. Initiation: Define project, feasibility, charter.

2. Planning: Scope, WBS, schedule (CPM/PERT), budget, resources.

3. Execution: Direct and manage work, team development.

4. Monitoring & Controlling: Track progress, manage changes, control quality/schedule/cost.

5. Closing: Formal acceptance, handover, lessons learned.

E. Heuristic & Meta-Heuristic Algorithms

• Heuristics: Problem-specific "rule-of-thumb." Fast, gives good but not optimal solution. E.g., greedy algorithm for knapsack.

• Meta-Heuristics: High-level framework for exploring large solution spaces. General-purpose, often stochastic. Examples:

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

  • Simulated Annealing (SA): Mimics annealing process, accepts worse solutions to escape local optima.

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

VI. SUPPLY CHAIN MANAGEMENT (SCM) PRINCIPLES

A. Bull-Whip Effect

• Definition: Demand distortion (amplification of order variability) as one moves upstream from retailer to manufacturer.

• Causes:

  1. Demand Forecasting: Using past orders to forecast creates amplification.

  2. Order Batching: Large, infrequent orders cause spikes.

  3. Price Fluctuations: Forward buying during promotions.

  4. Rationing & Gaming: Shortage -> over-ordering, then cancellation.

• Mitigation Strategies:

  • Information Sharing: Point-of-Sale (POS) data sharing (e.g., VMI).

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

  • Lead Time Reduction: Shorter lead times reduce need for safety stock.

  • Eliminate incentives for forward buying/gaming.

• Uses: Understanding supply chain dynamics, improving coordination, reducing costs.

B. Logistics Management

• Inbound Logistics: Activities from suppliers to manufacturing (procurement, receiving, internal material handling). Importance: Major cost component (40-50% of logistics cost), impacts production continuity.

• Outbound Logistics: Activities from manufacturing to customer (finished goods storage, distribution, delivery). Importance: Directly impacts customer satisfaction, market reach, and competitive advantage.

• Role of Inventory in Logistics: Acts as buffer against uncertainty in supply/demand. Trade-off: higher inventory = better service but higher holding cost. Optimal level balances cost vs. service.

C. Flow in SCM

D. Evolution: MRP → MRP II → ERP → SCM

• MRP (Material Requirements Planning): Dependent demand items. Time-phased netting of requirements from BOM and MPS. Focus: Material planning.

• MRP II (Manufacturing Resource Planning): Extends MRP to include capacity planning (CRP), shop floor control, and financials. Closed-loop system.

• ERP (Enterprise Resource Planning): Integrates all business functions (finance, HR, sales, manufacturing) across the enterprise. Single database.

• SCM (Supply Chain Management): Extends integration beyond firm to suppliers and customers. Focus on collaborative planning, visibility, and optimization of the entire chain.

E. Cross-Docking

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

• Importance:

  • Reduces inventory holding costs and space.

  • Reduces handling and storage time → faster throughput.

  • Improves product freshness (perishables).

• Disadvantages:

  • Requires excellent coordination, synchronization, and information systems.

  • High infrastructure investment (docking facilities, sorting systems).

  • Vulnerable to delays; no buffer inventory.

F. Outsourcing in SCM

• Definition: Contracting non-core activities (e.g., logistics, IT, manufacturing) to third-party specialists (3PL, 4PL).

• Importance:

  • Allows firm to focus on core competencies.

  • Access to specialized expertise and technology.

  • Potential for cost reduction (economies of scale).

  • Converts fixed costs to variable costs, increases flexibility.

G. SCM and E-Business Linkage

• E-Business (E-commerce): Use of internet for business transactions.

• Linkage to SCM:

  • E-Procurement: Online purchasing, auctions, catalogs → streamlines sourcing.

  • Online Ordering: Direct customer orders → real-time demand signal.

  • Digital Supply Chains: End-to-end visibility, real-time tracking, collaborative planning portals.

  • Global Reach: Enables sourcing and sales globally, but increases complexity.

  • Data-Driven: Rich data for forecasting and analytics.

H. Expenditure and Opportunities in SCM

• Major Expenditure Areas:

  • Transportation: Largest cost (40-50%).

  • Inventory: Carrying costs (capital, storage, obsolescence).

  • Warehousing: Fixed costs of facilities.

  • Information Technology: Systems (ERP, WMS, TMS).

  • Administration & Labor.

• Opportunities for Improvement:

  • Cost Reduction: Via optimization, outsourcing, process re-engineering.

  • Service Improvement: Faster delivery, higher fill-rates, customization.

  • Risk Mitigation: Diversification, visibility, contingency planning.

  • Sustainability: Green logistics, carbon footprint reduction.

VII. DECISION ANALYSIS AND GAME THEORY

A. Basic Concepts

• Players: Decision-makers (Row player, Column player).

• Strategies: Options available to each player.

• Payoff Matrix: Table showing outcomes (payoffs) for each strategy combination.

B. Pure vs. Mixed Strategies

• Pure Strategy: Player chooses one specific strategy deterministically.

• Mixed Strategy: Player chooses strategies according to a probability distribution (e.g., play A with 0.6, B with 0.4). Used when no pure strategy equilibrium exists.

C. Dominance Rule

• Dominant Strategy: Strategy that yields a higher payoff for a player regardless of what the opponent does.

  • For Row player: $$\displaystyle a_{ij} > a_{kj} $$ for all j (strict dominance).

• Use: Can eliminate dominated strategies to simplify game matrix.

D. Basic Assumptions

1. Rationality: Players aim to maximize their own payoff.

2. Common Knowledge: All players know the payoff matrix and rules.

3. Fixed Rules: Game structure and strategies are fixed and known.

VIII. ADVANCED OPTIMIZATION APPROACHES (Brief Overview)

A. Heuristic Algorithms

• Definition: "Rule-of-thumb" procedures that guide search for good solutions quickly.

• Characteristics: Problem-specific, fast, no guarantee of optimality, often greedy.

• Example: Nearest Neighbor for TSP, greedy for knapsack.

B. Meta-Heuristic Algorithms

• Definition: High-level, problem-independent frameworks that guide heuristics to explore large solution spaces effectively.

• Characteristics: Stochastic, can escape local optima, adaptable.

• Examples:

  • Genetic Algorithms (GA): Population-based, uses selection, crossover, mutation.

  • Simulated Annealing (SA): Single-solution based, probabilistic acceptance of worse moves.

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

  • Ant Colony Optimization (ACO): Inspired by ant foraging behavior (pheromone trails).

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EXAM DISTINCTION: Heuristic = specific rule. Meta-heuristic = general framework that uses heuristics as components. Both are used for NP-hard problems where exact methods are too slow.