Operations Management¶
Why This Matters¶
Operations management is the engine room of every business. While strategy decides what to do and finance decides how to fund it, operations decides how to actually deliver it -- reliably, efficiently, and at scale. Every euro of revenue passes through an operations system before it becomes profit. A restaurant with brilliant marketing still fails if the kitchen cannot serve food on time. A factory with cutting-edge products still bleeds cash if its inventory ties up working capital needlessly.
This course builds the quantitative toolkit to diagnose, measure, and improve any operations system -- whether it is a hospital emergency room, a pizza delivery chain, a luxury goods manufacturer, or an airline. The core question is always: How do we deliver the right product/service, at the right time, at the right cost, at the right quality?
How It All Connects¶
Operations Management sits at the intersection of every other MBA discipline:
- Financial Accounting -- Inventory appears on the balance sheet as a current asset; WIP and finished goods directly affect working capital and cash flow ratios
- Managerial Accounting -- Cost systems (variable vs. absorption), make-vs-buy decisions, and variance analysis all depend on understanding process capacity and utilization
- Marketing Management -- Demand forecasts feed capacity planning; service quality (wait times, stockouts) shapes customer satisfaction and retention
- Corporate Finance -- Capacity investments are capital budgeting decisions (NPV of a new production line); inventory financing affects WACC through working capital needs
- Operations Strategy (S3) -- This course provides the analytical foundation; Operations Strategy builds on it to address competitive positioning through operations (trade-offs, focus, operational excellence)
The Big Picture Flow:
Demand Forecast --> Capacity Planning --> Process Design --> Flow Management
(Marketing) (CapEx/Finance) (This Course) (This Course)
|
+----------+----------+
| | |
Throughput Inventory Queues
Analysis Management Analysis
Lesson 1: Introduction -- Efficiency vs. Effectiveness¶
Case: Benihana
Core Concept¶
Operations management is NOT just about cutting costs. It is about aligning the operations system with the business strategy. Two fundamental dimensions:
- Efficiency = doing things right (low cost, high utilization, minimal waste)
- Effectiveness = doing the right things (meeting customer needs, quality, speed)
The tension between these two drives most operational decisions.
The Operations Pipeline Analogy¶
Think of any operations system as a pipeline through which items (products, customers, information) flow. The pipeline has three elements:
| Element | Definition | Example (Benihana) |
|---|---|---|
| Items | Things being processed | Customers/diners |
| Activities | Basic units of work that add value | Seating, cooking, serving, payment |
| Processors/Servers | Resources that perform activities | Chefs, tables, bar area |
Five Competitive Dimensions of Operations¶
| Dimension | What It Measures | Key Parameters |
|---|---|---|
| Capacity | Revenue-generating ability | Throughput, max throughput |
| Flexibility | Variety of items + ability to scale up/down | Product mix impact, idle capacity |
| Agility | Speed of response | Throughput time |
| Efficiency | Investment and cost per unit of output | WIP, utilization, labor efficiency |
| Quality | Ability to meet specifications | Defect rate, rework cost |
Benihana Insight¶
Benihana brilliantly aligns efficiency with effectiveness: the teppanyaki grill is both entertainment (effectiveness -- the customer experience) and an operations system that maximizes table turns, minimizes food waste, and reduces kitchen space (efficiency). The chef IS the server AND the show.
Cross-reference: This efficiency-vs-effectiveness tension reappears in Operations Strategy (S3) as the productivity frontier and strategic trade-offs.
Lesson 2: Process Analysis -- Introduction¶
Case: Pizzas Dani
Core Formulas¶
Throughput (T): The quantity of items actually processed per unit of time.
T = items completed / time period
Cycle Time (CT): The average interval between successive completions (inverse of throughput).
CT = 1 / T
If a barber serves 3 customers/hour, CT = 20 minutes.
Maximum Throughput = Capacity:
Capacity = Processor Availability (hours/day) / Processor Consumption per item (hours/item)
Throughput Time (TT): Total time an item spends in the system (entry to exit), including all waits, processing, and transport.
Work-in-Process (WIP): Items that have entered but not yet exited the system.
Little's Law -- The Universal Operations Equation¶
WIP = T × TT
| Variable | Meaning | Units |
|---|---|---|
| WIP | Average work-in-process | items |
| T (or TH) | Average throughput | items/time |
| TT | Average throughput time | time |
Intuition: Imagine a highway. If 60 cars enter per hour (T) and each takes 0.5 hours to cross (TT), there are always 30 cars on the highway (WIP). More cars on the road (higher WIP) at the same entry rate means each car takes longer (higher TT).
Back-of-Napkin: Little's Law¶
- If 10 people are in a system and throughput is 2/hour, average time = 10/2 = 5 hours
- A restaurant has 60 seats, average meal = 1.5 hours --> max throughput = 60/1.5 = 40 diners/hour
- A factory has 500 units of WIP and produces 100 units/day --> average throughput time = 5 days
Variable Definitions¶
| Variable | Name | Units |
|---|---|---|
| T (TH) | Throughput | items/time |
| CT | Cycle time | time/item |
| TT | Throughput time | time |
| WIP | Work-in-process | items |
Work Content and Labor Efficiency¶
Work content = total processor time actually spent producing one item (the real value-added time).
Labor content = work content attributed to human labor only (excludes machine-only time).
Labor Efficiency = Total labor content of items produced / Total labor availability (hours paid)
Quick Mental Math: If 2 hairdressers work 8 hours each (960 min total) and serve 20 customers at 27 min labor content each (540 min), labor efficiency = 540/960 = 56.25%. The rest is idle time or non-value-added work.
Lesson 3: Process Analysis -- Product Mix¶
Case: Arlanzones
The Product Mix Problem¶
When a system processes multiple types of items, capacity depends on the mix. A hairdresser takes 20 min for men, 30 min for women. With a 50/50 mix:
Average processing time = (20 + 30) / 2 = 25 min = 0.417 hours
Max throughput = 8 hours / 0.417 hours = 19.2 customers/day
Capacity Analysis: The 3-Step Method¶
| Step | Action | Formula |
|---|---|---|
| 1 | Consumption: Define item mix, determine processor time per item | Weighted average if multiple products |
| 2 | Availability: Hours available per processor × number of processors | Total availability = individual × count |
| 3 | Capacity: Divide availability by consumption | Capacity = Availability / Consumption |
Bottleneck Principle¶
For a given mix of items, the maximum throughput of a system is constrained by the processor with the lowest capacity. That processor is the bottleneck.
The bottleneck is always a processor, not an activity. One processor may perform multiple activities.
Utilization¶
Utilization = Actual Throughput / Processor Capacity = (Consumption × Demand) / Processor Availability
Key insight: In a multi-processor system, the bottleneck runs at 100% utilization. All other processors are underutilized. If ALL processors ran at 100%, items would pile up infinitely in front of the bottleneck.
Product Mix Shifting the Bottleneck¶
The bottleneck can change when the product mix changes. A processor that is the bottleneck for one mix may NOT be the bottleneck for a different mix.
Implication for managers: Before investing in capacity (buying machines, hiring workers), always check whether the bottleneck will remain the same under realistic demand scenarios.
Cross-reference: Product mix decisions connect to Managerial Accounting (contribution margin per unit of bottleneck resource) and Marketing Management (which products to promote).
Lesson 4: Process Analysis -- Process Improvement¶
Case: Privalia
Removing Bottlenecks¶
Once the bottleneck is identified, there are several levers:
- Add capacity at the bottleneck (more processors of same type)
- Reduce processing time at the bottleneck (training, technology, process redesign)
- Offload activities from the bottleneck to non-bottleneck processors (task reallocation)
- Reduce variability in processing times
- Reduce setup/changeover time (increases effective capacity)
Crucial Insight: Whack-a-Mole¶
As soon as one bottleneck is removed, a new bottleneck appears elsewhere in the system. Capacity improvement is an ongoing process of eliminating bottlenecks, one after another.
Example: Adding a second assembly worker (bottleneck at 16 chairs/day) increases assembly capacity to 32/day, but the SYSTEM capacity only rises to 20/day because varnishing (20/day) becomes the new bottleneck. Doubling bottleneck capacity does NOT double system capacity.
Effect of Batch Size on Capacity¶
When activities have setup time (preparation independent of batch size):
Capacity = M / (p + s/Q)
Where M = available time, p = unit processing time, s = setup time, Q = batch size. As Q increases, the setup time per unit (s/Q) shrinks, and capacity approaches M/p asymptotically.
Minimum batch size to achieve required capacity C:
Qmin = C / [(M − C × p) / s]
If Qmin is negative, the required capacity is unachievable regardless of batch size.
Back-of-Napkin: Batch Sizing¶
- Setup = 60 min, processing = 1 min/unit, 8-hour day (480 min)
- Batch of 1: capacity = 480/61 = ~8 units/day
- Batch of 60: capacity = 480/(1 + 60/60) = 480/2 = 240 units/day
- Batch of 480: capacity = 480/(1 + 60/480) = ~437 units/day (approaching max of 480)
Cross-reference: Process improvement connects to Corporate Finance -- capacity investments are NPV/IRR decisions. Every bottleneck removal has a cost-benefit trade-off.
Lesson 5: Input/Output Analysis -- Introduction¶
From Static to Dynamic Analysis¶
Process analysis tells us whether capacity fits demand on average. But even when average capacity > average demand, predictable fluctuations (time-of-day, day-of-week, seasonality) create temporary imbalances that cause waiting.
The Input/Output Curve¶
Place two observers: Observer A counts cumulative arrivals over time. Observer B counts cumulative departures (completions). Plot both on the same time axis.
Reading the I/O curve: - Vertical gap at any time = number of items waiting (queue length / inventory) - Horizontal gap at any cumulative count = waiting time for that item - Area between curves = total waiting done by all items (units: item-hours)
Key KPIs from I/O Analysis¶
Avg. queue length = Area between curves / Total time period observed
Avg. waiting time = Area between curves / Total number of items
Worked Example: Ski Lift¶
Lift capacity: 3,000 skiers/hour. Arrivals vary: - Morning (9:00-12:30): 3,600/hr --> excess of 600/hr - Lunch (12:30-14:00): 1,600/hr --> surplus capacity of 1,400/hr - Afternoon (14:00-18:00): 3,200/hr --> excess of 200/hr
Queue builds during morning at 600 skiers/hr for 3.5 hours = 2,100 skiers max queue at 12:30.
Maximum waiting time (for skier arriving at 12:30): Cumulative arrivals by 12:30 = 3,600 × 3.5 = 12,600 Time for lift to process 12,600 skiers = 12,600 / 3,000 = 4.2 hours Wait = 4.2 − 3.5 = 0.7 hours = 42 minutes
Total waiting (area of triangle) = 1/2 × 2,100 × (3.5 + 1.5) = 5,250 skier-hours
Intuition: The Bathtub¶
Think of a bathtub: water flows in (arrivals) and drains out (processing). If inflow > outflow, the water level (queue) rises. Even if total daily inflow = total daily outflow, timing mismatches create temporary buildup.
Why This Matters¶
I/O curves let managers prototype operational changes before committing resources: test different staffing levels, operating hours, pricing policies, or capacity upgrades by simply redrawing the curves.
Lesson 6: Input/Output -- Services¶
Case: Vall d'Hebron Hospital
Service-Specific Challenges¶
Services differ from manufacturing in I/O analysis: - Customers ARE the items -- they experience the wait directly - Cannot inventory service -- unused capacity in one period is lost forever - Simultaneity -- production and consumption happen at the same time
Managing Predictable Demand Peaks¶
| Strategy | Mechanism | Example |
|---|---|---|
| Shift demand | Appointments, dynamic pricing, off-peak incentives | Hospital scheduling elective surgeries |
| Flex capacity | Part-time staff, overtime, cross-training | Hospital calling in extra nurses for Monday mornings |
| Absorb with buffer | Waiting rooms, virtual queues | Emergency room triage |
Hospital Application¶
Emergency rooms face both predictable patterns (Monday mornings, Friday nights) and unpredictable surges. I/O analysis handles the predictable part; queueing theory (Lessons 11-12) handles the random part.
Lesson 7: Input/Output -- Manufacturing¶
Case: Poma de Tuixent
Manufacturing I/O Differences¶
- Items can be inventoried (buffer between input and output)
- Setup times and batch sizes create lumpy output patterns
- Multiple product types create sequencing decisions
Production Smoothing¶
When demand is seasonal but capacity is fixed, managers face a trade-off: - Chase strategy: Adjust capacity to match demand (hire/fire, overtime) -- higher labor costs, lower inventory - Level strategy: Produce at constant rate, build inventory during slack periods -- higher inventory costs, stable workforce
This directly sets up Lesson 8 on Aggregate Planning.
Lesson 8: Aggregate Planning¶
Case: Athena Luxury Purses
The Aggregate Planning Problem¶
Given a demand forecast over multiple periods, decide how much to produce each period by choosing among:
| Decision Variable | Cost Driver |
|---|---|
| Regular production | Labor cost per unit |
| Overtime production | Premium labor cost (typically 1.5x) |
| Subcontracting | Higher unit cost but no fixed overhead |
| Hiring workers | Hiring/training cost |
| Firing/laying off workers | Severance, morale, legal costs |
| Building inventory | Holding cost (storage, capital, obsolescence) |
| Backorders/lost sales | Penalty cost, customer dissatisfaction |
Cost Trade-offs Framework¶
The objective is to minimize total cost across all periods subject to meeting demand. This is fundamentally a trade-off between:
Capacity adjustment costs (hiring, firing, overtime) vs. Inventory holding costs vs. Stockout/backorder costs
Solving Approaches¶
- Spreadsheet enumeration: Try different strategies (pure chase, pure level, mixed), calculate total cost for each
- Linear programming: Formalize as optimization problem (often too complex for exam, but conceptually important)
Key Intuition¶
- If hiring/firing is cheap relative to holding: use chase strategy
- If holding is cheap relative to hiring/firing: use level strategy
- If overtime premium is small: use overtime before hiring
- Subcontracting makes sense when demand spikes are temporary and unpredictable
Back-of-Napkin: Aggregate Planning¶
Ask three questions: 1. What is total demand over the planning horizon? 2. What is total regular-time capacity? 3. What is the gap? --> This must be filled by overtime, subcontracting, inventory buildup, or lost sales
Cross-reference: Aggregate planning connects to Corporate Finance (working capital management) and Managerial Accounting (cost behavior -- fixed vs. variable, relevant costs for decisions).
Lesson 9: Productivity Management¶
Case: Surgikos
Productivity = Output / Input¶
Productivity = Throughput (units or revenue) / Resources consumed (labor, capital, materials)
Partial vs. Total Factor Productivity¶
| Measure | Formula | Use |
|---|---|---|
| Labor productivity | Output / Labor hours | Compare across shifts, plants |
| Capital productivity | Output / Capital employed | Assess equipment ROI |
| Total factor productivity | Output / (weighted sum of all inputs) | Overall efficiency benchmark |
Productivity Improvement Levers¶
- Process improvement -- reduce waste, eliminate non-value-added steps
- Technology -- automate, better equipment
- Workforce management -- training, incentives, job design
- Quality improvement -- fewer defects = less rework = more effective output
- Capacity utilization -- spreading fixed costs over more units
The Surgikos Insight¶
Productivity is not just about working harder or faster. It is about working smarter -- redesigning processes, removing bottlenecks, and ensuring that every unit of input creates maximum output.
Lesson 10: Midterm Exam -- covers Lessons 1-9
Lesson 11: Queue Management -- Introduction¶
Why Queues Form Even When Capacity > Demand¶
Process analysis and I/O curves handle predictable imbalances. But even when average capacity exceeds average demand, random variability in arrivals or service times creates queues. Five customers might arrive in the first 5 minutes of an hour, then none for 10 minutes.
Anatomy of a Queueing System¶
Three defining elements:
1. Arrivals:
| Parameter | Symbol | Meaning |
|---|---|---|
| Mean inter-arrival time | tₐ | Average time between arrivals |
| Arrival rate | lambda (λ) | λ = 1/tₐ (items/time) |
| Std. dev. of inter-arrival time | σₐ | Variability of arrivals |
| Coefficient of variation | CVₐ | CVₐ = σₐ / tₐ |
2. Service:
| Parameter | Symbol | Meaning |
|---|---|---|
| Mean service time | tₛ | Average time to serve one item |
| Service rate | mu (μ) | μ = 1/tₛ (items/time per server) |
| Std. dev. of service time | σₛ | Variability of service |
| Coefficient of variation | CVₛ | CVₛ = σₛ / tₛ |
3. Design:
| Parameter | Symbol | Meaning |
|---|---|---|
| Number of servers | S | Parallel identical servers |
| Queue discipline | -- | FIFO, priority, etc. |
| Queue structure | -- | Pooled (single line) vs. separate lines |
Utilization¶
ρ = λ / (S × μ)
| Variable | Meaning |
|---|---|
| ρ (rho) | Utilization (fraction of time servers are busy) |
| λ (lambda) | Arrival rate |
| μ (mu) | Service rate per server |
| S | Number of servers |
Critical thresholds: - ρ < 1: System is stable; queues form but eventually clear - ρ = 1: No slack; any fluctuation causes infinite queue buildup - ρ > 1: System is fundamentally under-capacity; queue grows without bound
Coefficient of Variation (CV) -- Measuring Variability¶
| CV Value | Interpretation |
|---|---|
| CV = 0 | Deterministic (perfectly predictable, like a metronome) |
| CV = 1 | Typical of random/Poisson processes (customers arriving independently) |
| CV > 1 | Highly variable, chaotic, unpredictable |
Four Key KPIs¶
| KPI | Symbol | What It Measures |
|---|---|---|
| Avg. number waiting in queue | Lq | Queue length |
| Avg. waiting time in queue | Wq | Time before service starts |
| Avg. number in system | L | Queue + being served |
| Avg. total time in system | W | Waiting + service time |
Little's Law in Queueing Notation¶
L = λ × W (system level) Lq = λ × Wq (queue level)
Relationships Between KPIs¶
W = Wq + tₛ (total time = wait + service) L = Lq + Lₛ (total in system = in queue + in service) Lₛ = S × ρ = λ × tₛ (avg. number being served)
Once you know ANY ONE of {Lq, Wq, L, W}, you can derive ALL the others.
The Sakasegawa Approximation¶
Lq ≈ [ρ^√(2(S+1)) / (1 − ρ)] × [(CVₐ² + CVₛ²) / 2]
| Component | What It Does |
|---|---|
| ρ^√(2(S+1)) / (1 − ρ) | Captures utilization and server count effect |
| (CVₐ² + CVₛ²) / 2 | Captures variability effect |
Simplified Special Case (S=1, CVₐ = CVₛ = 1)¶
Lq = ρ² / (1 − ρ)
The Utilization Trap -- Back-of-Napkin¶
| Utilization (ρ) | Lq (S=1, CV=1) | Interpretation |
|---|---|---|
| 50% | 0.5 | Short queue |
| 70% | 1.6 | Manageable |
| 80% | 3.2 | Getting long |
| 85% | 4.8 | Expect queues to explode past here |
| 90% | 8.1 | Very long waits |
| 95% | 18.1 | Unacceptable |
| 99% | 98.0 | System is broken |
Rule of thumb: If utilization > 85%, expect queues to explode. The relationship is NONLINEAR -- going from 90% to 95% roughly doubles the queue.
Intuition: The Traffic Jam¶
A highway at 50% capacity flows freely. At 80% capacity, any small disruption (a lane change, a slow driver) causes a ripple. At 95% capacity, the highway locks up. The same physics governs any queueing system.
Lesson 12: Queue Management -- Pooling¶
Case: Etihad Airways
Three Managerial Levers¶
1. Add capacity (increase S or μ) - Directly reduces ρ - Cost: more servers, equipment, labor
2. Reduce variability (lower CVₐ or CVₛ) - Arrival variability: appointments, reservations, demand smoothing, metered entry - Service variability: standardize procedures, train for consistency, segment customers by complexity - Effect: as powerful as adding capacity, often cheaper
3. Pool resources (combine separate queues into one) - Instead of 2 separate queues with 1 server each, use 1 pooled queue with 2 servers - Variability evens out across the larger system - Dramatically reduces probability that one server starves while another is overwhelmed
Pooling Effect -- Why It Works¶
With separate queues: if one server finishes and has no customers, that capacity is wasted even though the other queue may be long.
With a pooled queue: idle servers automatically take the next customer from the shared queue. No capacity is wasted.
Quantitative impact: Pooling reduces Lq through BOTH the utilization term (same ρ but more servers) and the √(2(S+1)) exponent in the Sakasegawa formula.
Trade-offs of Pooling¶
- May feel less personal (one long line vs. "your" dedicated server)
- Requires cross-trained staff (broader skills, possibly higher wages)
- Perception management: customers may perceive one long line as worse than multiple short lines, even if the math says otherwise
Etihad Airways Application¶
Airlines pool check-in counters (economy, business, first) vs. dedicate them. Pooling improves throughput and reduces worst-case waits, but business/first-class passengers expect dedicated service. The solution: partial pooling -- dedicated lines for premium, pooled for economy.
Lesson 13: Production Planning¶
Case: Edentel
From Aggregate to Detailed Planning¶
Aggregate planning (Lesson 8) decides how much to produce each period. Production planning decides what specific products to produce, when, and in what sequence.
Key Decisions¶
- Lot sizing: How much of each product per production run? (connects to EOQ in Lesson 14)
- Sequencing: In what order to produce different products? (minimize total setup time)
- Scheduling: When exactly to start and finish each job? (Gantt charts)
MRP Logic (Material Requirements Planning)¶
Net requirement = Gross requirement − On-hand inventory − Scheduled receipts
Work backward from finished goods delivery dates to determine when to start each activity, accounting for lead times at each stage.
Lesson 14: Inventory -- Batching (EOQ)¶
Case: MORSA
Why Batch? Three Effects¶
- Increases capacity (fewer setups = more production time)
- Decreases unit cost (fixed setup cost spread over more units)
- Increases inventory (larger batches = more average stock)
The EOQ Model¶
Setup: Demand is stable at D units/year. Each order costs S to place. Holding one unit for one year costs H = v × i (unit value × carrying cost rate).
Two competing costs:
Ordering cost/year = S × D/Q (decreases with batch size Q)
Holding cost/year = H × Q/2 (increases with batch size Q)
The optimal batch size (Economic Order Quantity):
EOQ = √(2 × D × S / H)
| Variable | Name | Units |
|---|---|---|
| D | Annual demand | units/year |
| S | Setup/ordering cost | $/order |
| H | Holding cost per unit per year | $/unit/year |
| v | Unit value | $/unit |
| i | Carrying cost rate | %/year |
| Q | Order quantity (batch size) | units |
At EOQ, ordering cost = holding cost (the two curves cross at the minimum of total cost).
Back-of-Napkin: EOQ Square Root Relationship¶
The EOQ formula has a square root, which makes it extremely robust:
- EOQ doubles when demand quadruples (because √4 = 2)
- A 40% error in demand estimation causes only ~18% error in EOQ
- A 7% rounding of EOQ (e.g., to box sizes) causes < 1% increase in total cost
This is the most important practical insight about EOQ: Don't obsess over getting perfect data. The formula is forgiving.
When EOQ Breaks Down¶
The simple EOQ formula assumes: - Only ordering and holding costs matter - No quantity discounts - Constant, known demand - No capacity constraints
When other costs exist (quantity discounts, free shipping above a threshold), use a spreadsheet approach: enumerate total costs for candidate batch sizes and pick the minimum.
Batch Size and Capacity¶
Capacity = M / (p + s/Q)
Where M = available time, p = unit processing time, s = setup time, Q = batch size.
Minimum batch for required capacity C:
Qmin = C / [(M − C × p) / s]
Cross-reference: Inventory on the balance sheet (Financial Accounting) -- larger batches mean higher average inventory, higher current assets, lower inventory turnover ratio. The CFO cares about EOQ because it directly affects working capital.
Lesson 15: Inventory -- Safety Stock¶
Case: MORSA (continued)
Why Safety Stock?¶
EOQ assumes demand is perfectly known. In reality, demand fluctuates. Safety stock is the buffer inventory kept to protect against stockouts caused by demand uncertainty.
The Safety Stock Formula¶
SS = z × σ_d × √VP
| Variable | Name | Meaning |
|---|---|---|
| SS | Safety stock | Extra units held as buffer |
| z | Safety factor | From normal distribution table (higher z = lower stockout risk) |
| σ_d | Std. dev. of demand per period | Measures demand uncertainty |
| VP | Vulnerable period | Number of periods the safety stock must cover |
Vulnerable Period¶
VP = LT + RP
| Component | Meaning |
|---|---|
| LT (Lead Time) | Time from placing order to receiving goods |
| RP (Review Period) | How often inventory is checked (0 if continuous review) |
Common z Values¶
| Service Level | Stockout Risk | z Value |
|---|---|---|
| 84.1% | 15.9% | 1.00 |
| 90.0% | 10.0% | 1.28 |
| 95.0% | 5.0% | 1.64 |
| 97.7% | 2.3% | 2.00 |
| 99.0% | 1.0% | 2.33 |
| 99.9% | 0.1% | 3.09 |
Key Intuition: The Square Root Effect¶
Safety stock grows with the square root of the vulnerable period, NOT linearly.
- Doubling the lead time does NOT double the safety stock
- Safety stock for 4 periods = 2x safety stock for 1 period (because √4 = 2)
- Safety stock for 9 periods = 3x safety stock for 1 period
Why? Because demand fluctuations in different periods are independent and partially cancel out. A high-demand day is likely followed by a normal or low-demand day.
Reorder Point¶
ROP = (Average demand per period × VP) + SS
When inventory drops to ROP, place an order.
Back-of-Napkin: Safety Stock¶
- Demand: μ = 100/day, σ = 20/day, lead time = 9 days, continuous review (RP=0)
- For 95% service level: z = 1.64
- SS = 1.64 × 20 × √9 = 1.64 × 20 × 3 = 98.4 units
- ROP = 100 × 9 + 98.4 = 998.4 units --> order when inventory hits ~1,000 units
Important Nuance: Use Forecast Error, Not Demand Variability¶
If you have a good forecasting model, the safety stock should protect against forecast error (the residual uncertainty), not total demand variability. Better forecasts --> smaller σ --> less safety stock needed.
Cross-reference: Safety stock ties to Financial Accounting (inventory valuation methods -- FIFO, LIFO, weighted average) and to Marketing Management (service level targets are ultimately customer-facing promises).
Lesson 16: Inventory -- Perishable Goods (Critical Fractile)¶
Case: Fiore di Zucca
The Newsvendor Problem¶
For products with a short life cycle (fashion, perishable food, newspapers, seasonal toys), unsold inventory has little or no value after the selling period. You must decide how much to produce/order before knowing demand.
Two Key Costs¶
| Cost | Symbol | Definition |
|---|---|---|
| Cost of underage | Cᵤ (or G) | Profit lost per unit if demand > supply (you could have sold it) |
| Cost of overage | Cₒ (or L) | Loss per unit if supply > demand (unsold unit you must dispose of) |
The Critical Fractile Formula¶
Fc = Cᵤ / (Cᵤ + Cₒ) = G / (G + L)
Decision rule: Produce quantity q* such that:
P(Demand ≤ q*) = Fc
Graphically: compute Fc, draw a horizontal line at that level on the cumulative demand distribution, and read off q* where it intersects.
How to Apply¶
- Calculate G: How much more you earn if you produce one more unit and sell it
- G = selling price - unit cost (or: incremental margin)
- Calculate L: How much you lose if you produce one more unit and DON'T sell it
- L = unit cost - salvage value
- Compute Fc = G / (G + L)
- Read q* from the CDF of demand at probability Fc
Worked Example: Book Publisher¶
- Cost to produce: 9 euro/book
- Selling price: 30 euro/book
- G = 30 - 9 = 21 euro
- L = 9 - 0 = 9 euro (no salvage)
- Fc = 21 / (21 + 9) = 0.70
- From the CDF: P(D <= 1,100) = 0.70
- Optimal print run: 1,100 books
Worked Example: Airline Revenue Management¶
- Business class ticket: 2,100 euro; Economy: 700 euro
- Unlimited economy demand
- G = 2,100 - 700 = 1,400 (extra revenue from business vs. economy)
- L = 700 (empty seat that could have been economy)
- Fc = 1,400 / (1,400 + 700) = 0.667
- From CDF of business demand: optimal seats to protect = 27
Key Intuition¶
- High margin, low disposal cost (G >> L) --> Fc close to 1 --> produce aggressively (you make a lot per sale, lose little per unsold unit)
- Low margin, high disposal cost (G << L) --> Fc close to 0 --> produce conservatively (each unsold unit is very costly)
- Equal costs (G = L) --> Fc = 0.5 --> produce the median demand
Back-of-Napkin: Critical Fractile¶
- If you make 3x more selling a unit than you lose not selling it: Fc = 3/(3+1) = 75% --> produce the 75th percentile of demand
- If you make equal amounts: Fc = 50% --> produce the median
- If you lose 2x more from overproduction than you gain from a sale: Fc = 1/(1+2) = 33% --> produce conservatively
Computing Expected Profit¶
The critical fractile tells you the optimal quantity but NOT the expected profit. To find expected profit: 1. Discretize the demand distribution into branches (e.g., 5 equal-probability branches using quantiles) 2. Calculate profit for each demand branch given the optimal production quantity 3. Weight by probability and sum
Lesson 17: Supply Chain Coordination¶
The Bullwhip Effect¶
Small fluctuations in consumer demand get amplified as they propagate upstream through the supply chain. A 5% increase in retail demand might translate to a 40% spike in orders at the manufacturer.
Causes¶
- Demand signal processing: Each level forecasts based on orders received (not actual consumer demand)
- Order batching: EOQ logic creates lumpy orders
- Price fluctuations: Forward-buying during promotions
- Rationing/shortage gaming: Over-ordering when supply is scarce
Mitigation Strategies¶
- Information sharing: Share point-of-sale data upstream (e.g., Walmart-P&G partnership)
- Vendor-managed inventory (VMI): Supplier manages retailer's inventory
- Everyday low pricing (EDLP): Reduces forward-buying
- Smaller, more frequent batches: Reduces order lumpiness (requires reducing setup costs)
- Collaborative forecasting: Joint demand planning across supply chain tiers
Cross-reference: Supply chain design connects to Operations Strategy (S3 -- "What Is the Right Supply Chain for Your Product?" by Fisher) and Marketing Management (distribution channel strategy).
Lessons 18-20: Operations Day (simulation) -- applies all concepts from the course in a competitive team simulation
Lesson 21: Integration -- Organizational Issues¶
Case: Bioco
Operations and Organization Design¶
Operations decisions are not made in a vacuum. They interact with: - Incentive systems -- workers optimizing their own metrics (utilization) may hurt system performance (throughput time) - Organizational silos -- marketing promises delivery dates that operations cannot meet - Culture -- continuous improvement requires psychological safety to report problems
Key Tensions¶
| Tension | Operations View | Other Department View |
|---|---|---|
| Utilization vs. responsiveness | Keep machines busy | Deliver fast |
| Batch size | Large batches = efficient | Small batches = flexible |
| Inventory | Costly buffer | Sales safety net |
| Quality | Process control | Design innovation |
Lesson 22: Integration -- Service Excellence¶
Case: Etnia Barcelona
Service Operations Framework¶
Service excellence requires aligning: 1. Service concept -- what value are you delivering? 2. Service delivery system -- the operations pipeline for services 3. Demand management -- matching variable demand to fixed capacity 4. Customer management -- shaping expectations, managing waits
Perceived vs. Actual Waiting Time¶
Research shows that perceived wait time matters more than actual wait time. Occupied time feels shorter than unoccupied time. Uncertain waits feel longer than known waits. Unexplained waits feel longer than explained waits.
Operations as Competitive Advantage¶
When operations and strategy are perfectly aligned, operations becomes the source of sustained competitive advantage -- not just a cost center. This is the bridge to Operations Strategy (S3).
Lesson 23: Final Exam -- comprehensive, covers all course material
Quick Reference¶
Master Formula Sheet¶
| # | Formula | Variables | When to Use |
|---|---|---|---|
| 1 | CT = 1/T | CT = cycle time, T = throughput | Convert between throughput and cycle time |
| 2 | WIP = T × TT | WIP = work-in-process, TT = throughput time | Little's Law -- universal relationship |
| 3 | Capacity = Availability / Consumption | hours/day divided by hours/item | Capacity of a single processor |
| 4 | ρ = λ / (S × μ) | ρ = utilization, λ = arrival rate, μ = service rate, S = servers | Server utilization in queueing |
| 5 | Lq ≈ [ρ^√(2(S+1)) / (1 − ρ)] × [(CVₐ² + CVₛ²)/2] | Sakasegawa approximation | Average queue length |
| 6 | L = λ × W | L = avg in system, W = avg time in system | Little's Law for queues |
| 7 | Lq = λ × Wq | Lq = avg in queue, Wq = avg wait in queue | Little's Law for queue portion |
| 8 | W = Wq + tₛ | Total time = wait + service | Linking queue and system times |
| 9 | Lₛ = S × ρ | Avg number in service | From utilization |
| 10 | EOQ = √(2DS/H) | D = demand, S = setup cost, H = holding cost | Optimal batch/order size |
| 11 | SS = z × σ_d × √VP | z = service factor, σ_d = demand std dev, VP = vulnerable period | Safety stock calculation |
| 12 | VP = LT + RP | LT = lead time, RP = review period | Vulnerable period |
| 13 | ROP = (μ_d × VP) + SS | μ_d = avg demand/period | When to reorder |
| 14 | Fc = Cᵤ / (Cᵤ + Cₒ) = G / (G + L) | Cᵤ = underage cost, Cₒ = overage cost | Critical fractile -- newsvendor |
| 15 | Capacity (with batches) = M / (p + s/Q) | M = available time, p = unit time, s = setup, Q = batch | Capacity with setup times |
| 16 | Qmin = C / [(M − C×p)/s] | C = required capacity | Minimum batch for target capacity |
| 17 | Avg wait (I/O) = Area / Number of items | Area between cumulative curves | Input/output analysis |
| 18 | Avg queue (I/O) = Area / Time period | Area between cumulative curves | Input/output analysis |
Quick Mental Math Shortcuts¶
| Situation | Shortcut |
|---|---|
| Little's Law | WIP / Throughput = Time. Always. |
| Utilization > 85% | Queues will explode nonlinearly |
| EOQ sensitivity | Demand x4 --> EOQ x2 (square root) |
| Safety stock scaling | Lead time x4 --> SS x2 (square root) |
| Critical fractile = 0.5 | Produce the median demand |
| Fc > 0.5 | High-margin product -- produce above median |
| Fc < 0.5 | Low-margin/high-disposal-cost -- produce below median |
| Bottleneck removed | A new one always appears elsewhere |
| Capacity = Availability/Consumption | Use weighted avg consumption for product mix |
| I/O area = triangle | 1/2 × base × height for cumulative curve gaps |
Glossary¶
| Term | Definition |
|---|---|
| Aggregate Planning | Deciding production levels across multiple periods to balance capacity costs, inventory costs, and demand |
| Availability | Total time a processor has available for work in a given period |
| Batch | A group of items processed together, sharing a single setup |
| Bottleneck | The processor with the lowest capacity in a system; determines system capacity |
| Bullwhip Effect | Amplification of demand variability as it moves upstream in a supply chain |
| Capacity | Maximum throughput of a processor or system (items/time) |
| Chase Strategy | Adjusting production rate each period to match demand (variable workforce) |
| Coefficient of Variation (CV) | Standard deviation divided by mean; unitless measure of relative variability |
| Critical Fractile (Fc) | The ratio Cᵤ/(Cᵤ + Cₒ) that determines optimal production quantity for perishable goods |
| Cycle Time (CT) | Average time between successive completions; CT = 1/Throughput |
| EOQ | Economic Order Quantity; the batch size that minimizes total ordering + holding cost |
| Holding Cost (H) | Cost of keeping one unit in inventory for one year (includes capital, storage, obsolescence) |
| Input/Output Curve | Cumulative arrival and departure functions plotted on same time axis |
| Items | Things that flow through and are processed by the operations system |
| Labor Efficiency | Ratio of labor content of output to total labor hours available |
| Lead Time (LT) | Time from placing an order to receiving goods |
| Level Strategy | Producing at a constant rate regardless of demand fluctuations |
| Little's Law | WIP = Throughput × Throughput Time; holds for any stable system |
| Newsvendor Problem | Single-period inventory decision for perishable/seasonal goods |
| Pooling | Combining separate queues/resources to share variability and improve performance |
| Processors/Servers | Resources (people, machines, equipment) that perform activities on items |
| Reorder Point (ROP) | Inventory level at which a new order should be placed |
| Review Period (RP) | Time between consecutive inventory checks |
| Safety Stock (SS) | Extra inventory held to protect against demand uncertainty during the vulnerable period |
| Sakasegawa Approximation | Formula to estimate average queue length as a function of utilization, servers, and variability |
| Setup Time | Time to prepare a processor for a new batch (independent of batch size) |
| Throughput (T) | Actual rate of items processed per unit of time |
| Throughput Time (TT) | Total time an item spends in the system from entry to exit |
| Utilization (ρ) | Fraction of available capacity actually used; ρ = λ / (S × μ) |
| Vulnerable Period (VP) | Time window that safety stock must cover: VP = LT + RP |
| WIP (Work-in-Process) | Items that have entered but not yet exited the system |
| Work Content | Total processor time consumed in producing one item |