Systems Engineering (ME-703 (C)) - Unit 3 Short Notes

UNIT 3: Operations Research & Supply Chain Management

I. Linear Programming (LP)

Problem Formulation: Convert word problems into standard mathematical form:

• Objective Function: Maximize or Minimize $$\displaystyle Z = c_1x_1 + c_2x_2 + ... + c_nx_n $$

• Constraints: $$\displaystyle a_{11}x_1 + a_{12}x_2 + ... + a_{1n}x_n (\le, =, \ge) b_1 $$

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

Simplex Method (Maximization with ≤ constraints):

1. Convert to Standard Form: Add slack variables $$\displaystyle s_i \ge 0 $$ to convert ≤ to equality.

$$a_{11}x_1 + ... + s_1 = b_1$$

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

3. Simplex Tableau: Include objective row ($$\displaystyle Z - c_jx_j = 0 $$) and constraint rows.

4. Optimality Test: If all coefficients in objective row (reduced costs) are ≤ 0 (for max), current solution is optimal.

5. Iteration:

   • Entering Variable: Most positive coefficient in objective row.

   • Leaving Variable: Minimum positive ratio (RHS / pivot column element) → determines pivot row.

   • Pivot Operation: Update tableau using row operations to make pivot element = 1, others in column = 0.

6. Read Solution: Values of basic variables from RHS column; non-basic = 0.

[!TIP] Common Pitfalls:

• Forgetting to convert maximization to minimization for simplex (or vice versa) when using big-M/two-phase.

• Ratio test: only consider positive elements in pivot column; zero/negative ignored.

• Optimality condition differs for minimization (all coefficients ≥ 0).

II. Transportation Problems

Problem Formulation:

• Balanced: Total supply = Total demand.

• Unbalanced: Add dummy row/column with zero cost to balance.

• Objective: Minimize total transportation cost.

Initial Basic Feasible Solution (IBFS):

[!TIP] VAM is often near-optimal; use for quick IBFS before MODI/Stepping Stone.

Degeneracy:

• Definition: IBFS has fewer than $(m+n-1)$ positive allocations (where $m$=rows, $n$=cols).

• Cause: Simultaneous exhaustion of supply & demand, or zero-cost cells.

• Resolution:

  • Epsilon (ε) Method: Assign tiny value ε (< any positive cost) to zero allocation to make it positive temporarily for MODI calculations.

  • Alternative: Allocate arbitrarily small positive value (e.g., 0.001) to degenerated cell.

Transportation with Penalties:

• For unfulfilled demand, add dummy destination with cost = original cost + penalty.

• Solve balanced problem; dummy allocations indicate unmet demand.

Optimality Testing:

• MODI Method (preferred for degeneracy):

  1. Compute $$\displaystyle u_i, v_j $$ using $$\displaystyle u_i + v_j = c_{ij} $$ for basic cells.

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

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

  4. If any $$\displaystyle \Delta_{ij} < 0 $$, select most negative for next allocation; form loop; adjust allocations (+θ, -θ).

• Stepping Stone Method: Trace closed loop for each non-basic cell; compute net change in cost.

III. Supply Chain Management (SCM) Fundamentals

Core Flows:

Logistics in SCM:

• Inbound Logistics: Receiving, storing, handling incoming materials from suppliers.

  • Importance: Ensures smooth production; reduces inventory costs; improves supplier relationships.

• Outbound Logistics: Storing, handling, distributing finished goods to customers.

  • Importance: Directly impacts customer satisfaction; reduces delivery time/cost; enables competitive advantage.

Bull-Whip Effect:

• Definition: Demand variability amplifies as orders move upstream (retailer → distributor → manufacturer).

• Causes:

  1. Demand forecast updating.

  2. Order batching.

  3. Price fluctuations (discounts).

  4. Rationing & gaming (shortage anticipation).

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

• Mitigation Strategies:

  • Vendor Managed Inventory (VMI)

  • Continuous Replenishment

  • Sharing point-of-sale (POS) data

  • Stabilizing prices (everyday low pricing)

  • Reducing lead times

• Uses/Implications: Highlights need for information sharing; drives CPFR (Collaborative Planning, Forecasting, Replenishment).

SCM Strategies & Practices:

• Cross-Docking:

  • Definition: Unloading inbound trucks/containers directly to outbound vehicles with minimal storage.

  • Importance: Reduces inventory holding costs & handling; speeds up distribution.

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

• Outsourcing:

  • Importance: Focus on core competencies; reduce costs; access expertise; scalability; risk sharing.

SCM Evolution & Integration:

• MRP (Material Requirements Planning): Material planning for manufacturing (dependent demand).

• MRP II (Manufacturing Resource Planning): Integrated MRP with capacity planning, shop floor control.

• ERP (Enterprise Resource Planning): Integrated across all business functions (finance, HR, SCM).

• SCM: Extends beyond firm to network coordination; focuses on flows across supply chain.

• Link with E-Business: E-procurement, e-marketplaces, online order tracking, e-fulfillment; enables real-time information sharing.

Expenditure & Opportunities:

• Expenditure: Major costs in transportation, inventory, warehousing, order processing.

• Opportunities: Cost reduction via optimization, just-in-time (JIT), lean logistics, technology (IoT, blockchain), sustainability (green SCM).

Role of Inventory:

• Buffer against demand/supply uncertainty.

• Decoupling point between stages.

• Economies of scale in ordering/production.

• But increases holding costs, risk of obsolescence. Optimal balance via EOQ, safety stock.

IV. Inventory Management

Economic Order Quantity (EOQ) Model: Assumptions:

• Constant, known demand rate ($D$).

• Instantaneous replenishment (lead time = 0 or constant).

• No shortages allowed.

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

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

Basic Formula:

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

• $D$ = Annual demand (units/year)

• $S$ = Ordering cost (Rs/order)

• $H$ = Holding cost (Rs/unit/year)

Derived Metrics:

• Number of orders per year: $$\displaystyle N = \frac{D}{Q^*} $$

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

• Total Annual Cost (TAC): $$\displaystyle TAC = \frac{D}{Q}S + \frac{Q}{2}H + PD $$ (P = unit cost, last term constant)

• Minimum TAC: $$\displaystyle TAC_{\min} = \sqrt{2DSH} + PD $$

[!TIP] Unit Consistency: Ensure $D$, $S$, $H$ have same time unit (usually yearly). If holding cost given as % of unit cost: $$\displaystyle H = i \times P $$, where $i$ = carrying rate (e.g., 0.08/year).

EOQ with Quantity Discounts:

• All-Units Discount: Discount applies to all units if order quantity ≥ breakpoint.

  • Compute EOQ at each price level (using discounted $P$ for $$\displaystyle H = i \times P $$).

  • If EOQ feasible (≥ breakpoint), compute TAC.

  • If EOQ < breakpoint, use breakpoint quantity for TAC.

  • Choose quantity with lowest TAC among feasible EOQs and breakpoints.

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

  • More complex; compute TAC for each price range separately.

EOQ with Infinite Shortage Cost (No shortages allowed):

• Same as basic EOQ; assumption already prohibits shortages.

Inventory Classification:

[!TIP] ABC = Pareto principle (80/20 rule). VED = importance-based (used in maintenance, spares).

V. Queuing Theory

Basic Concepts:

• Queue: Customers waiting for service.

• Queue Discipline: Order of service (FCFS, LCFS, priority, random).

• Arrival Process: Often Poisson (random, independent arrivals).

• Service Process: Often Exponential (memoryless, constant mean rate).

• System Capacity: Finite/infinite waiting space.

• Channels: Single/multiple servers.

Poisson Distribution (Arrivals):

Probability of exactly $k$ arrivals in time $t$:

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

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

• Mean = Variance = $\lambda t$.

Exponential Distribution (Service Times):

Probability service time exceeds $t$:

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

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

• Mean service time = $1/\mu$.

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

Single-Server Queue (M/M/1):

• Arrivals: Poisson ($\lambda$)

• Service: Exponential ($\mu$)

• Capacity: Infinite

• Discipline: FCFS

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

• Performance Measures:

  • Average number in system: $$\displaystyle L_s = \frac{\rho}{1-\rho} $$

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

  • Average time in system: $$\displaystyle W_s = \frac{1}{\mu - \lambda} $$

  • Average time in queue: $$\displaystyle W_q = \frac{\lambda}{\mu(\mu - \lambda)} $$

[!TIP] Past paper example: "20 customers served per hour" → $$\displaystyle \mu = 20 $$/hr. "More than 15 minutes" → $$\displaystyle t = 0.25 $$ hr. $$\displaystyle P(T > 0.25) = e^{-20 \times 0.25} = e^{-5} \approx 0.0067 $$.

VI. Project Management (PERT/CPM)

Network Diagrams:

• AON (Activity-on-Node): Nodes = activities; arrows = dependencies (preferred).

• AOA (Activity-on-Arrow): Arrows = activities; nodes = events.

• Dummy Activity: Zero duration; used to show dependency without work (preserves logic in AOA).

Critical Path Identification:

1. Forward Pass (ES, EF):

   • $$\displaystyle ES = \max(EF \text{ of predecessors}) $$

   • $$\displaystyle EF = ES + \text{duration} $$

2. Backward Pass (LF, LS):

   • $$\displaystyle LF = \min(LS \text{ of successors}) $$

   • $$\displaystyle LS = LF - \text{duration} $$

3. Float:

   • Total Float (TF): $LS - ES$ or $LF - EF$ (slack without delaying project).

   • Free Float (FF): $ES \text{ of next activity} - EF \text{ of current}$ (slack without delaying successors).

4. Critical Path: Path with zero total float; longest path; determines project duration.

[!TIP] Critical path can change during project; monitor near-critical paths.

PERT Time Estimates (Probabilistic):

• Optimistic (O): Minimum time if everything goes well.

• Pessimistic (P): Maximum time if major delays.

• Most Likely (M): Normal time under typical conditions.

• Expected Time: $$\displaystyle \mu = \frac{O + 4M + P}{6} $$

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

Project Completion Probability:

1. Compute critical path with PERT times.

2. Project Mean ($$\displaystyle T_\mu $$) = sum of expected times on critical path.

3. Project Variance ($$\displaystyle T_{\sigma^2} $$) = sum of variances on critical path.

4. For due date $D$, compute Z-score:

$$Z = \frac{D - T_\mu}{\sqrt{T_{\sigma^2}}}$$

5. Use standard normal table: $$\displaystyle P(T \le D) = \Phi(Z) $$.

   • If $Z$ negative: $$\displaystyle P(T \le D) = 1 - \Phi(|Z|) $$.

[!TIP] Only critical path activities affect project variance; non-critical paths have slack.

PERT vs CPM:

Applications of PERT/CPM:

• Scheduling complex projects.

• Resource allocation.

• Crashing (time-cost trade-off).

• Monitoring progress (EVT).

Phases of Project Management:

1. Initiation – Define scope, objectives.

2. Planning – WBS, scheduling, budgeting, risk planning.

3. Execution – Coordinate resources, implement plan.

4. Monitoring & Controlling – Track progress, manage changes.

5. Closure – Deliverables, lessons learned.

Heuristic & Meta-Heuristic Algorithms:

• Heuristics: Rule-of-thumb for quick, good solutions (e.g., nearest neighbor for TSP).

• Meta-Heuristics: Higher-level frameworks (e.g., Genetic Algorithms, Simulated Annealing, Tabu Search) for complex optimization; escape local optima.

Network Logics:

• Finish-to-Start (FS): Successor starts after predecessor finishes (most common).

• Start-to-Start (SS): Successor starts after predecessor starts.

• Finish-to-Finish (FF): Successor finishes after predecessor finishes.

• Start-to-Finish (SF): Rare; successor finishes after predecessor starts.

VII. Game Theory (Limited Coverage)

Basic Assumptions:

• Finite number of players ($n$).

• Each player has finite strategies.

• Players choose strategies independently (simultaneously or sequentially with known moves).

• Payoffs known to all.

• Rationality: Players maximize their own payoff.

Strategies:

• Pure Strategy: Specific choice (e.g., always choose A).

• Mixed Strategy: Probability distribution over pure strategies.

Dominance Rule:

• Strict Dominance: Strategy A dominates B if payoff(A) > payoff(B) for all opponent strategies.

• Weak Dominance: payoff(A) ≥ payoff(B) for all, and > for at least one.

• Elimination: Dominated strategies can be removed iteratively to simplify game matrix.

[!TIP] Used in 2-player zero-sum games to reduce strategy sets before solving via simplex or graphical method.

Key Formulas Summary: