Systems Engineering (ME-703 (C)) - Unit 1 Short Notes

UNIT 1: OPERATIONS RESEARCH & SUPPLY CHAIN MANAGEMENT

I. LINEAR PROGRAMMING (LP)

Problem Formulation:

• Identify decision variables (e.g., \(x_1, x_2\) = units to produce).

• Define objective function (Maximize profit \(Z\) or Minimize cost).

• List constraints as linear inequalities (resource limits: labor, materials, capacity).

• Include non-negativity constraints: \(x_i \ge 0\).

[!TIP] Common Pitfall: Forgetting non-negativity or misinterpreting "≤" vs. "≥" from word problems.

Simplex Method (Maximization with ≤ constraints):

1. Convert to Standard Form:

   • Add slack variables (\(s_i \ge 0\)) to ≤ constraints to turn them into equations.

   • Example: \(3x_1 + 2x_2 \le 6\) becomes \(3x_1 + 2x_2 + s_1 = 6\).

2. Initial Basic Feasible Solution (IBFS): Set decision variables \(x_i = 0\). Slack variables take the RHS values.

3. Set up Initial Simplex Tableau: Columns for all variables (decision + slack). Rows for constraints + objective (with \(Z\) row).

4. Iteration Steps:

   • Pivot Column: In \(Z\) row (bottom), select the most negative coefficient (entering variable).

   • Pivot Row: Compute ratio = (RHS value) / (positive pivot column entry). Choose smallest non-negative ratio (leaving variable).

   • Pivot Operation: Make pivot element = 1 (divide row). Use row operations to make all other entries in pivot column = 0.

5. Optimality: Stop when all coefficients in Z row are ≥ 0. Optimal solution is the RHS values of basic variables.

6. Interpretation:

   • Slack Variable Value > 0: Corresponding constraint is not binding (resource has surplus).

   • Slack Variable Value = 0: Constraint is binding (resource fully used).

   • Surplus Variable: Used for ≥ constraints (not in current past papers).

Special Cases:

• Alternative Optimal Solutions: Zero coefficient in Z row for a non-basic variable at optimum.

• Unbounded Solution: All entries in pivot column ≤ 0 when a negative Z coefficient exists.

• Infeasible Solution: Artificial variable (\(a_i\)) remains positive in final solution (Big-M or Two-Phase method).

II. TRANSPORTATION PROBLEMS

Objective: Minimize total transportation cost from \(m\) origins (sources) to \(n\) destinations.

Initial Basic Feasible Solution (IBFS) Methods:

Degeneracy:

• Definition: An IBFS is degenerate if the number of allocated cells (positive allocations) is less than \(m + n - 1\).

• Cause: Simultaneous satisfaction of a row and column during allocation.

• Resolution (Epsilon Method):

  1. Assign a very small positive value (\(\epsilon > 0\)) to one of the zero-cell allocations in the degenerate cell.

  2. Treat \(\epsilon\) as a regular allocation for MODI optimality test.

  3. Proceed with MODI. Final solution ignores \(\epsilon\).

Unbalanced Problems:

• Condition: \(\sum \text{Supply} \neq \sum \text{Demand}\).

• Solution: Introduce a Dummy Row (if supply < demand) or Dummy Column (if supply > demand).

  • Cost = 0 for genuine routes.

  • Penalty Cost for unfulfilled requirement: Add to dummy row/column cells. This penalty represents the cost of not meeting demand/supply.

Optimality Testing (MODI / u-v Method):

1. For IBFS with \(m+n-1\) allocations, compute dual variables \(u_i\) (for rows) and \(v_j\) (for columns) using: \(u_i + v_j = c_{ij}\) for all basic cells.

   • Set \(u_1 = 0\) (arbitrary), solve for others.

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

3. Check Optimality:

   • All \(\Delta_{ij} \ge 0\) → Optimal solution.

   • Any \(\Delta_{ij} < 0\) → Not optimal. Select most negative \(\Delta_{ij}\) for next allocation.

4. Revised Allocation: Draw a closed loop from the entering cell (most negative \(\Delta_{ij}\)). Alternate + and - at occupied cells. Adjust allocations: subtract min(allocations at - cells) from - cells, add to + cells.

III. INVENTORY MANAGEMENT

Economic Order Quantity (EOQ) Model - Basic Assumptions:

• Demand rate (\(D\)) is constant & known.

• Instantaneous replenishment (order arrives all at once).

• No shortages allowed.

• Fixed ordering cost (\(C_o\)) per order.

• Constant holding/carrying cost (\(C_h\)) per unit per year.

EOQ Formula & Derivations:

Total Annual Cost = Ordering Cost + Holding Cost + (Purchase Cost, if variable)

\[ > TC = \frac{D}{Q} C_o + \frac{Q}{2} C_h + D \cdot C_u > \]

Where \(Q\) = order quantity, \(C_u\) = unit cost.

Optimal Order Quantity (\(Q^*\)):

$$Q^* = \sqrt{\frac{2 D C_o}{C_h}}$$

\boxed{Q^* = \sqrt{\frac{2 D C_o}{C_h}}}

Derived Parameters:

• Number of Orders per Year: \(N^* = D / Q^*\)

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

• Minimum Total Cost (excluding purchase): \(TC_{\min} = \sqrt{2 D C_o C_h}\)

[!TIP] EOQ is the point where ordering cost = holding cost.

Variations:

1. Finite Production Rate (EPQ):

   • Production rate (\(P\)) > demand rate (\(D\)).

   • Inventory builds gradually.

   • Formula: \(Q^* = \sqrt{\frac{2 D C_o}{C_h \left(1 - \frac{D}{P}\right)}}\)

2. Quantity Discounts:

   • All-or-Nothing: Must buy entire batch at discounted price if \(Q \ge\) breakpoint.

   • Incremental: Discount applies only to units above breakpoint.

   • Procedure: Calculate EOQ at each price. If feasible (within price break range), compute TC. Compare TCs at EOQ and at each breakpoint's minimum \(Q\). Choose minimum TC.

3. Shortages Allowed (Backordering):

   • Shortage cost per unit per year (\(C_s\)) is finite.

   • Optimal \(Q^*\) and maximum shortage (\(S^*\)) formulas exist. If \(C_s \to \infty\), solution reduces to basic EOQ (no shortages).

Inventory Classification Systems:

IV. SUPPLY CHAIN MANAGEMENT (SCM) CORE CONCEPTS

Flows in SCM:

1. Material Flow: Physical movement of goods from supplier → manufacturer → distributor → customer.

2. Information Flow: Orders, forecasts, schedules, acknowledgments flowing both upstream & downstream.

3. Money/Cash Flow: Payments, credit, consignment, flowing opposite to material flow (customer → distributor → manufacturer → supplier).

Logistics in SCM:

• Inbound Logistics: Activities from supplier to manufacturer (sourcing, procurement, inbound transportation, receiving, storage). Goal: Efficient, cost-effective material inflow.

• Outbound Logistics: Activities from manufacturer to customer (finished goods storage, order processing, outbound transportation, delivery). Goal: Timely, accurate, low-cost product delivery.

• Reverse Logistics: Handling returns, repairs, recycling, disposal (part of outbound).

[!TIP] Inbound focuses on cost reduction (procurement), Outbound on service level (customer satisfaction).

Role of Inventory in Logistics:

• Decoupling: Separates stages (production vs. demand) to absorb variability.

• Buffer: Against uncertainties in demand, supply, lead time.

• Economies: Enables bulk purchasing/production, reduces transportation costs.

• Cost vs. Service Trade-off: Higher inventory → better service but higher holding cost.

SCM Expenditures & Opportunities:

• Major Expenditures: Procurement (60-70%), Transportation (10-20%), Inventory Holding (10-30%).

• Opportunities: Reduce total cost (not just unit cost), improve asset utilization (inventory turns), enhance customer service (fill rate, lead time).

Outsourcing in SCM:

• Strategic Role: Focus on core competencies (e.g., design, marketing), outsource non-core (logistics, IT, manufacturing).

• Importance: Access to expertise, scalability, cost reduction, risk sharing.

• Key Partner: 3PL (Third-Party Logistics) providers for transportation, warehousing.

V. KEY SCM PHENOMENA & STRATEGIES

Bull-Whip Effect:

• Definition: Amplification of demand variability as one moves upstream in the supply chain (from retailer to manufacturer to supplier).

• Causes:

  1. Demand Signal Processing (forecasting based on orders, not sales).

  2. Order Batching (periodic large orders).

  3. Price Fluctuations (forward buying).

  4. Rationing & Gaming (shortage-induced over-ordering).

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

• Mitigation Strategies:

  • Information Sharing: Point-of-Sale (POS) data sharing (CPFR).

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

  • Eliminate incentives for forward buying (everyday low pricing).

  • Reduce lead times and order batching (smaller, frequent orders).

Cross-Docking:

• Definition: Logistics practice where inbound shipments are directly transferred to outbound transportation with minimal or no storage. Goods "cross" the dock.

• Process:

  DiagramCANVAS: Inbound trucks unload → sorting/consolidation area → outbound trucks load. Little to no rack storage.

• Advantages:

  • Drastically reduced inventory holding costs & space.

  • Reduced material handling & labor.

  • Shorter lead times.

  • Faster product turnover.

• Disadvantages:

  • Requires high coordination & precise scheduling.

  • Needs advanced IT systems (WMS, TMS).

  • High risk of disruption (no buffer stock).

  • Not suitable for all products (needs high, predictable volume).

Evolution: MRP → ERP → SCM Integration:

1. MRP (Material Requirements Planning): Internal focus. Explodes master production schedule into component requirements using BOM & inventory records. Closed-loop MRP includes feedback.

2. MRP II (Manufacturing Resource Planning): Extends MRP to include capacity planning, shop floor control, finance.

3. ERP (Enterprise Resource Planning): Integrates all business functions (finance, HR, sales, manufacturing, supply chain) into a single database. Real-time data across the enterprise.

4. ERP & SCM: Modern ERP systems have SCM modules (procurement, logistics, demand planning). True SCM extends beyond a single firm to the network of partners.

SCM & e-Business:

• Linkage: e-Business (B2B, B2C) is a key enabler for SCM.

• Integration: E-commerce platforms generate demand signals; e-procurement automates purchasing; e-logistics enables tracking; collaborative portals connect partners.

• Impact: Reduces transaction costs, improves information flow, enables mass customization, creates new distribution channels (direct-to-consumer).

VI. QUEUING THEORY

Basic Concepts:

• Queue: Customers waiting for service.

• Queue Discipline: Rule for selecting next customer (FCFS most common; also LCFS, priority, random).

• Arrival Process: Pattern of customer arrivals. Often modeled as Poisson with rate \(\lambda\) (avg. arrivals per unit time).

• Service Mechanism: Number of servers (\(s\)), service time distribution. Often modeled as Exponential with rate \(\mu\) (avg. services per unit time per server).

• Population Size: Finite (limited customers) or Infinite.

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

Key Distributions & Calculations:

1. Exponential Service Time:

   • PDF: \(f(t) = \mu e^{-\mu t}\)

   • Probability service time > t: \(P(T > t) = e^{-\mu t}\)

   • Memoryless Property: \(P(T > s + t \| T > s) = P(T > t)\)

2. Poisson Arrivals:

   • PMF: \(P(N = n) = \frac{(\lambda t)^n e^{-\lambda t}}{n!}\)

   • Probability of exactly \(n\) arrivals in interval \(t\): Use formula above with \(\lambda t\) as mean.

Example (Past Paper): "20 customers served per hour" → \(\mu = 20\) per hour. "More than 15 minutes" → \(t = 0.25\) hours.

\[ P(T > 0.25) = e^{-20 \times 0.25} = e^{-5} \approx 0.0067 \]

VII. PROJECT MANAGEMENT (PERT/CPM)

PERT vs. CPM:

Network Analysis (Activity-on-Node - AON):

• Nodes: Represent activities.

• Arrows: Represent dependencies/precedence.

• Dummy Activity: Zero duration, used to preserve logic (show dependencies without time).

Forward Pass (Calculate ES, EF):

1. Start at Node 1: \(ES_1 = 0\).

2. For each node: \(EF_i = ES_i + t_i\).

3. For successor node \(j\): \(ES_j = \max(EF_i)\) over all immediate predecessors \(i\).

4. Project Duration (\(T_E\)) = EF of final node.

Backward Pass (Calculate LF, LS):

1. Start at Final Node: \(LF = T_E\).

2. For each node: \(LS_i = LF_i - t_i\).

3. For predecessor node \(j\): \(LF_j = \min(LS_i)\) over all immediate successors \(i\).

Slack / Total Float (TF):

\[ TF_i = LS_i - ES_i = LF_i - EF_i \]

• Critical Path: Path with TF = 0 for all activities. Determines project duration.

• Non-Critical Path: Activities with TF > 0.

PERT Time Estimates:

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

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

• Most Likely (\(M\)): Most realistic time.

• Expected Time (\(T_E\)):

  \[ T_E = \frac{O + 4M + P}{6} \]

• Variance (\(\sigma^2\)):

  \[ \sigma^2 = \left(\frac{P - O}{6}\right)^2 \]

Project Completion Probability:

1. Project Mean Duration (\(T_E\)) and Variance (\(\sigma_{cp}^2\)) from critical path (sum of variances along critical path).

2. For due date \(T_d\):

   \[ Z = \frac{T_d - T_E}{\sqrt{\sigma_{cp}^2}} \]

3. Find probability \(P(Z \leq \text{calculated Z})\) from standard normal table.

4. Probability of completion by \(T_d\) = \(P(Z \leq \text{calculated Z})\).

[!TIP] Variance along critical path is sum of individual variances (not standard deviations).

Heuristic & Meta-Heuristic Algorithms:

• Need: For large, complex scheduling problems where exact methods (like integer programming) are computationally intractable.

• Heuristics: Rule-based, quick, "good enough" solutions (e.g., Shortest Processing Time first, Earliest Due Date).

• Meta-Heuristics: Higher-level frameworks that guide heuristics to escape local optima.

  • Genetic Algorithms: Evolve a population of solutions via selection, crossover, mutation.

  • Simulated Annealing: Mimics cooling process; accepts worse solutions with decreasing probability.

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

• Application: Resource-constrained project scheduling, job shop scheduling, vehicle routing.

VIII. ADDITIONAL OR TOPICS

Game Theory:

• Scenario: Two or more decision-makers (players) in conflict/competition, each choosing strategies to maximize their own payoff.

• Assumptions:

  1. Rationality: Each player aims to maximize their payoff.

  2. Known Payoffs: All players know the payoff matrix for all strategy combinations.

  3. Competitive Situation: One player's gain is another's loss (zero-sum).

  4. Decisions are simultaneous or made without knowledge of the other's choice.

Strategies:

• Pure Strategy: Player chooses a specific, single strategy.

• Mixed Strategy: Player randomizes over strategies according to a probability distribution.

Dominance Rule (for Simplifying Payoff Matrix):

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

  • Action: Delete the dominated row \(j\).

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

  • Action: Delete the dominated column \(j\).

• Purpose: Reduces game size without losing the optimal solution.

Example: In a payoff matrix for Player A (rows), if every entry in Row 1 is greater than or equal to the corresponding entry in Row 2, Row 2 is dominated and can be removed.