Reliability Engineering (ME-703 (D)) - Unit 2 Short Notes

1. Linear Programming (LP) [HIGH FREQUENCY]

Formulation of LP Problems

• Objective: Maximize profit or minimize cost.

• Decision Variables: Represent quantities to be determined (e.g., \(x_1, x_2\)).

• Constraints: Represent limitations (resources, capacity, demand). Convert word problems into linear inequalities/equalities.

• Non-negativity: \(x_i \ge 0\) for all \(i\).

Simplex Method (Maximization with ≤ constraints)

• Step 1: Convert inequalities to equations by adding slack variables.

• Step 2: Form the initial simplex tableau. The basis consists of slack variables.

• Step 3: Optimality Test: Calculate \(C_j - Z_j\) for each variable.

  • If all \(C_j - Z_j \le 0\), current solution is optimal.

  • If any \(C_j - Z_j > 0\), the entering variable is the one with the most positive value.

• Step 4: Pivot Operation: Determine the leaving variable via Minimum Ratio Test (only positive entries in pivot column). Perform row operations to make pivot element = 1 and other elements in pivot column = 0.

• Step 5: Repeat Steps 3-4 until optimality is reached.

• Interpretation:

  • Solution: Values in the RHS column of final tableau for basic variables.

  • Shadow Price (Dual Value): Value in \(Z_j\) row under slack variable column; indicates marginal value of one additional unit of that resource.

  • Slack/Surplus: Value of slack/surplus variable in final solution (unused resource or excess).

[!TIP] Exam Focus: You will be asked to solve a 2- or 3-variable LP. Practice setting up the initial tableau and performing 2-3 iterations. Remember: \(Z_j = \sum (C_b \times a_{ij})\).

Special Cases

• Multiple Optimal Solutions: Occurs when a non-basic variable has \(C_j - Z_j = 0\) in the optimal tableau. Infinite solutions exist along the edge of the feasible region.

• Unbounded Solution: If the entering variable column has all \(\le 0\) entries, the problem is unbounded (objective can increase indefinitely).

2. Transportation Problems [HIGH FREQUENCY]

Initial Basic Feasible Solution (IBFS)

• North-West Corner (NWC) Rule:

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

  2. Allocate as much as possible: \(\min(\text{row supply}, \text{col demand})\).

  3. If row supply exhausted, move down; if col demand satisfied, move right.

  4. Repeat until all allocations are made.

• Vogel's Approximation Method (VAM) – Heavily Emphasized

  1. For each row and column, calculate penalty = difference between two smallest costs.

  2. Select the row/column with the highest penalty.

  3. In that row/column, allocate to the cell with the minimum cost.

  4. Adjust supply/demand, cross out exhausted row/col, recalculate penalties.

  5. Repeat until all allocations are made.

  • VAM generally yields a better starting solution (lower cost) than NWC.

Degeneracy

• Definition: Number of positive allocations in a BFS is less than \((m + n - 1)\), where \(m\) = rows, \(n\) = columns.

• Resolution: Introduce a very small allocation (\(\epsilon\)) in the zero-cost cell to make the number of allocations = \(m+n-1\). This \(\epsilon\) is treated as zero in cost calculations but ensures a full rank basis.

Unbalanced & Defective Transportation

• Unbalanced (Total Supply ≠ Total Demand):

  • If supply > demand: Add a dummy destination with zero transportation cost.

  • If demand > supply: Add a dummy source with zero transportation cost.

• Defective (Some demand/supply may be unfulfilled):

  • Introduce a dummy destination (for unfulfilled demand) or dummy source (for unused supply).

  • Assign a penalty cost (as given in problem) to these dummy cells.

  • The model now minimizes total transportation + penalty cost.

[!TIP] Common Pitfall: Forgetting to add dummy rows/columns for unbalanced problems, leading to an infeasible or incorrect solution.

3. Inventory Management [HIGH FREQUENCY]

Economic Order Quantity (EOQ) Model

• Assumptions:

  • Demand rate (\(D\)) is known, constant, and continuous.

  • Replenishment is instantaneous (order arrives in full at once).

  • No shortages allowed.

  • Ordering cost (\(S\)) per order is fixed.

  • Holding/carrying cost (\(H\)) per unit per year is constant.

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

  • Number of orders = \(D / Q\)

  • Ordering Cost = \((D/Q) \times S\)

  • Average Inventory = \(Q/2\)

  • Holding Cost = \((Q/2) \times H\)

  • Total Cost, TC(Q) = \(\frac{D}{Q}S + \frac{Q}{2}H + DC\) (where \(C\) = unit cost, constant).

EOQ Formulas

• Optimum Order Quantity:

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

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

• Optimum Time Between Orders (Cycle Time): \(T^* = Q^* / D\) (in years) or \(365/T^*\) days.

• Minimum Total Variable Cost (excluding purchase cost):

$$\boxed{TC_{min} = \sqrt{2DSH}}$$

EOQ with Quantity Discounts

• All-Units Discount: Discount applies to all units if order quantity \(Q\) is in a discount bracket.

• Incremental Discount: Discount applies only to units within the discount bracket.

• Procedure (All-Units):

  1. Calculate \(Q^*\) using base cost (no discount).

  2. If \(Q^*\) is not in a discount bracket, calculate TC at \(Q^*\) and at the minimum quantity of each discount bracket that is feasible.

  3. Choose the \(Q\) with the lowest total cost.

• Break-Even Discount Percentage: The discount % that makes the total cost at the discount price equal to the total cost at \(Q^*\).

[!TIP] Unit Conversion: Ensure \(D\), \(S\), \(H\) are in consistent time units (usually annual). If holding cost is given as a % of value (\(i\%\)), then \(H = i \times C / 100\).

4. Supply Chain Management (SCM) [HIGH FREQUENCY]

Bull-Whip Effect

• Definition: Phenomenon where order variability increases as we move upstream (from retailer to manufacturer) in the supply chain.

• Causes:

  • Demand Forecast Updating (using orders, not sales).

  • Order Batching (periodic ordering, promotions).

  • Price Fluctuations (forward buying).

  • Rationing & Gaming (shortage anticipation).

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

• Mitigation Strategies:

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

  • CPFR (Collaborative Planning, Forecasting, Replenishment): Shared data and joint planning.

  • Stabilize Prices (Everyday Low Pricing).

  • Order Smoothing / Eliminate Forward Buying.

Logistics

• Inbound Logistics: Activities from supplier to company (procurement, inbound transportation, receiving, storage, material handling). Focus: Efficient receipt and storage of raw materials.

• Outbound Logistics: Activities from company to customer (order processing, outbound transportation, distribution, final delivery). Focus: Timely, cost-effective delivery to end-user.

Key SCM Concepts

• Flows:

  • Material Flow: Physical movement of goods upstream/downstream.

  • Information Flow: Demand forecasts, orders, shipment notices (bidirectional).

  • Money Flow: Payments, credit, consignment (downstream).

• Cross-Docking:

  • Process: Inbound trucks unload directly to outbound trucks with minimal/zero storage.

  • Benefits: Reduced inventory holding, faster throughput, lower handling costs.

  • Disadvantages: Requires precise coordination, high IT investment, not suitable for all products.

• Outsourcing: Using third-party logistics (3PL) providers for logistics functions. Benefits: Focus on core competency, cost savings, expertise. Risks: Loss of control, dependency.

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 system for all business functions (finance, HR, SCM, CRM).

• Integrated SCM: Extends ERP visibility and coordination to external partners (suppliers, customers).

• Linkage with E-business: E-procurement (online purchasing), e-logistics (tracking), digital collaboration platforms (EDI, web portals).

5. Project Management [HIGH FREQUENCY]

Phases of Project Management

1. Initiation: Define project, stakeholders, high-level scope.

2. Planning: Develop scope, schedule (WBS, network), budget, resources, risk plan.

3. Execution: Coordinate people/resources, implement plans.

4. Monitoring & Controlling: Track progress, manage changes, ensure alignment with plan.

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

PERT vs. CPM

Network Analysis (Activity-on-Arrow - AOA)

• Activity: Task represented by an arrow.

• Event (Node): Start/end point of activities, represents a milestone.

• Dummy Activity: Zero duration, used to show logical dependencies without consuming time/resources.

• Forward Pass (Earliest Times):

  • \(E_1 = 0\)

  • \(E_j = \max(E_i + t_{ij})\) for all activities \((i,j)\) ending at event \(j\).

  • Earliest Finish of activity \((i,j)\): \(EF_{ij} = E_i + t_{ij}\)

• Backward Pass (Latest Times):

  • \(E_n = \text{Project Duration (from Forward Pass)}\)

  • \(L_i = \min(L_j - t_{ij})\) for all activities \((i,j)\) starting from event \(i\).

  • Latest Start of activity \((i,j)\): \(LS_{ij} = L_j - t_{ij}\)

• Float (Slack):

  • Total Float (TF): \(TF_{ij} = LS_{ij} - ES_{ij} = LF_{ij} - EF_{ij}\). Time an activity can be delayed without delaying project.

  • Free Float (FF): \(FF_{ij} = E_j - EF_{ij}\). Time an activity can be delayed without delaying early start of successors.

Critical Path

• Identification: Path from start to end with longest total duration. Activities on this path have TF = 0.

• Significance: Determines the minimum project completion time. Any delay on a critical activity delays the entire project. Focus of project control.

PERT Time Estimates & Variance

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

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

• Most Likely (M): Realistic time under normal conditions.

• Expected Activity Time:

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

• Activity Variance:

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

• Project Variance (σ²_p): Sum of variances (\(V\)) of activities on the critical path.

• Project Standard Deviation: \(\sigma_p = \sqrt{\sigma_p^2}\)

Project Completion Probability

• Standardization (Z-score):

$$\boxed{Z = \frac{T_d - TE_p}{\sigma_p}}$$

where \(T_d\) = due date, \(TE_p\) = expected project duration (sum of \(TE\) on critical path).

• Probability: \(P(\text{completion} \le T_d) = P(Z \le \text{calculated Z})\) from standard normal table.

• Probability of meeting a deadline: Look up cumulative probability for the calculated Z.

[!TIP] Critical Path in PERT: The critical path is the one with the longest expected time (TE), not necessarily the longest deterministic time. Variance is summed only for activities on this longest path.

6. Queuing Theory [HIGH FREQUENCY]

Basic Queueing System Components

• Arrival Process: Pattern of customer arrivals (often Poisson).

• Service Mechanism: Number of servers, service time distribution (often Exponential).

• Queue Discipline: Rule for selecting next customer for service (FIFO, LIFO, Priority).

• Queue Capacity: Finite or infinite.

• Population Size: Finite or infinite source.

Poisson Process for Arrivals

• Assumption: Arrivals are independent, average rate \(\lambda\) (arrivals/unit time) is constant.

• Probability of exactly \(k\) arrivals in time \(t\):

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

where \(\lambda t\) = average number of arrivals in interval \(t\).

Exponential Distribution for Service Times

• Probability Density Function (PDF): \(f(t) = \mu e^{-\mu t}\) for \(t \ge 0\).

• Mean Service Time: \(1/\mu\).

• Memoryless Property: \(P(T > s + t \mid T > s) = P(T > t)\). Past waiting doesn't affect future waiting time.

• Probability service time > \(t\):

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

Queue Disciplines

• FIFO (First-In-First-Out): Default, fair.

• LIFO (Last-In-First-Out): Stack-like.

• Priority: Customers assigned priority levels (e.g., emergency).

• Shortest Processing Time (SPT): Minimizes average waiting time.

• Random (SIRO): Service in random order.

[!TIP] Exam Problem: Given average service rate (e.g., 20 customers/hour), convert to per-minute rate: \(\mu = 20/60\). Then use \(P(T > t) = e^{-\mu t}\) for probability of service time exceeding \(t\) minutes.

7. Additional Operations Research Topics [MEDIUM/LOW FREQUENCY]

Game Theory

• Players: Decision-makers.

• Strategies: Alternatives available to a player.

• Payoff Matrix: Represents outcomes for each strategy combination.

• Pure Strategy: Optimal choice is definite (saddle point exists).

• Mixed Strategy: Player randomizes over strategies (no saddle point). Optimal solution found via dominance or simplex-like methods.

• Dominance Rule: A strategy \(A\) dominates strategy \(B\) if all payoffs in row \(A\) are ≥ corresponding payoffs in row \(B\) (for maximizer). Dominated strategy \(B\) can be eliminated.

• Basic Assumptions:

  1. Finite number of players/strategies.

  2. Payoffs are known and constant.

  3. Players choose rationally to maximize payoff.

  4. Decisions are made simultaneously (no knowledge of opponent's choice).

Inventory Classification Systems

• ABC Analysis (Based on Annual Consumption Value):

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

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

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

• VED Analysis (Based on Criticality/Urgency for Spare Parts):

  • V (Vital): No stock = production stoppage. Highest priority.

  • E (Essential): Important but not critical. Medium priority.

  • D (Desirable): Can be managed without. Lowest priority.

Advanced Optimization Algorithms

• Heuristic Algorithms: Problem-specific, rule-based, fast, provide good (but not necessarily optimal) solutions. Example: Nearest Neighbor for TSP.

• Meta-heuristic Algorithms: High-level frameworks that guide heuristics to escape local optima. Examples:

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

  • Simulated Annealing: Mimics physical annealing process.

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

Network Logic in Projects

• Predecessor: Activity that must finish before another can start.

• Successor: Activity that depends on a predecessor.

• Logical Relationships (FS, SS, FF, SF):

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

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

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

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