📐 PMP Math & Formulas — Part 2
Crashing, Agile, Probability, Resources, Procurement, Financial — With Interpretation Tables & Scenarios
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⚡ Crashing & Cost Slope
CrashingCrashing adds resources (people, equipment, overtime) to shorten schedule duration. It ALWAYS costs more money and is ONLY applied to critical path activities. The goal is maximum duration reduction for minimum additional cost. compresses schedule by adding resources. Only on critical path. Always costs more.
Cost Slope (cost per day saved)
Cost Slope = (Crash Cost - Normal Cost) / (Normal Duration - Crash Duration)
Lower cost slope = crash this activity first. It gives most time for least money.
Additional Crash Cost
Crash Cost Added = Cost Slope x Days Crashed
Multiply cost slope by number of days you are shortening the activity
New Project Cost
New Cost = Original Project Cost + Total Crash Cost Added
⭐ Crashing Interpretation Table
| Scenario | Status | What It Means | Action |
|---|---|---|---|
| Activity on critical path, low cost slope | CRASH FIRST ✓ | Cheapest way to save schedule days | Crash to maximum crash limit |
| Activity on critical path, high cost slope | CRASH LAST | Expensive option — use only if cheaper options exhausted | Crash only when no other option |
| Activity NOT on critical path | DO NOT CRASH ✗ | Crashing non-critical activities saves zero schedule time | Ignore for crashing purposes |
| Multiple critical paths after crashing | MUST CRASH ALL ✗ | Each critical path must be crashed simultaneously | Add cost slope of both activities |
| Crash limit reached on cheapest activity | MOVE TO NEXT | Activity fully crashed — move to next cheapest on critical path | Apply next lowest cost slope |
| Cost to crash > benefit of saving time | STOP CRASHING ✗ | Diminishing returns — not worth it | Consider fast-tracking instead |
🎯 Crashing Priority Order: 1) Find critical path. 2) Rank critical activities by cost slope (lowest first). 3) Crash cheapest first up to its crash limit. 4) Recheck if new critical paths formed. 5) Continue until target duration achieved.
📘 Crashing Scenarios — 4 Cases
Case 1 — Simple Crash to Save 3 Days
Critical path activities with cost slopes:
Activity A (critical): Normal=10d, Crash=7d, Cost slope=$1,500/day, Max crash=3 days
Activity B (critical): Normal=8d, Crash=6d, Cost slope=$2,200/day, Max crash=2 days
Activity C (critical): Normal=6d, Crash=5d, Cost slope=$3,500/day, Max crash=1 day
Need to save 3 days. Original cost = $200,000
Step 1: Crash A (cheapest at $1,500/day) x 3 days = +$4,500
New project cost = $200,000 + $4,500 = $204,500. Duration saved: 3 days. Done!
Activity A (critical): Normal=10d, Crash=7d, Cost slope=$1,500/day, Max crash=3 days
Activity B (critical): Normal=8d, Crash=6d, Cost slope=$2,200/day, Max crash=2 days
Activity C (critical): Normal=6d, Crash=5d, Cost slope=$3,500/day, Max crash=1 day
Need to save 3 days. Original cost = $200,000
Step 1: Crash A (cheapest at $1,500/day) x 3 days = +$4,500
New project cost = $200,000 + $4,500 = $204,500. Duration saved: 3 days. Done!
Case 2 — Crash Limit Reached, Must Continue to Next Activity
Need to save 5 days. Critical path: A(cost slope $800/day, max 2 days) | B($1,200/day, max 3 days) | C($2,000/day, max 2 days)
Step 1: Crash A x 2 days (max) = $800x2 = $1,600. Duration saved: 2 days. Need 3 more.
Step 2: Crash B x 3 days (max) = $1,200x3 = $3,600. Duration saved: 3 more. Total saved: 5.
Total crash cost = $1,600 + $3,600 = $5,200
Step 1: Crash A x 2 days (max) = $800x2 = $1,600. Duration saved: 2 days. Need 3 more.
Step 2: Crash B x 3 days (max) = $1,200x3 = $3,600. Duration saved: 3 more. Total saved: 5.
Total crash cost = $1,600 + $3,600 = $5,200
Case 3 — Two Critical Paths Emerge (Must Crash Both)
🎭 Exam Trap — Parallel Critical Paths:
After crashing Path 1 from 20 to 18 days, Path 2 (was 18 days) is now ALSO critical at 18 days.
To save 1 more day, must crash BOTH paths simultaneously:
Path 1 cheapest remaining: Activity D, cost slope $1,000/day
Path 2 cheapest: Activity E, cost slope $800/day
Cost to save 1 day = $1,000 + $800 = $1,800/day (combined cost slope)
After crashing Path 1 from 20 to 18 days, Path 2 (was 18 days) is now ALSO critical at 18 days.
To save 1 more day, must crash BOTH paths simultaneously:
Path 1 cheapest remaining: Activity D, cost slope $1,000/day
Path 2 cheapest: Activity E, cost slope $800/day
Cost to save 1 day = $1,000 + $800 = $1,800/day (combined cost slope)
Case 4 — Bridge Project: Find Cheapest Way to Meet Deadline
Current schedule = 30 days. Required = 25 days. Need to save 5 days.
Critical path activities:
Crash order (lowest cost slope first):
1) Curing x1 day = $400 | 2) Excavation x2 days = $1,800 | 3) Forming x2 days = $2,200
Total crash cost = $400+$1,800+$2,200 = $4,400 for 5 days saved
Critical path activities:
| Activity | Normal Duration | Crash Duration | Cost Slope | Max Crash |
|---|---|---|---|---|
| Excavation | 8 days | 6 days | $900/day | 2 days |
| Forming | 10 days | 7 days | $1,100/day | 3 days |
| Concrete Pour | 7 days | 6 days | $2,500/day | 1 day |
| Curing | 5 days | 4 days | $400/day | 1 day |
1) Curing x1 day = $400 | 2) Excavation x2 days = $1,800 | 3) Forming x2 days = $2,200
Total crash cost = $400+$1,800+$2,200 = $4,400 for 5 days saved
🚀 Fast-Tracking vs Crashing Comparison
| Factor | Crashing | Fast-Tracking |
|---|---|---|
| Method | Add more resources | Overlap sequential activities |
| Cost Impact | INCREASES cost significantly | Minimal direct cost increase |
| Risk Impact | Low additional risk | HIGH additional risk (rework) |
| Applicability | Any critical path activity | Only activities that CAN logically overlap |
| Best When | Budget available, risk tolerance low | Budget tight, risk tolerance higher |
| Exam Signal Word | "Add overtime" / "add resources" | "Overlap" / "start before predecessor done" |
Scenario — PM must save 4 days, has $10K budget:
Option A (Crash): Cost slope = $2,000/day x 4 days = $8,000 cost, no extra risk.
Option B (Fast-Track): Overlap Activity C into B — saves 4 days, $500 extra coordination cost but risks rework if B changes.
If budget is priority and risk is acceptable: Fast-Track. If risk is priority: Crash.
Option A (Crash): Cost slope = $2,000/day x 4 days = $8,000 cost, no extra risk.
Option B (Fast-Track): Overlap Activity C into B — saves 4 days, $500 extra coordination cost but risks rework if B changes.
If budget is priority and risk is acceptable: Fast-Track. If risk is priority: Crash.
🏃 Agile — Velocity & Throughput
Average Velocity
Velocity = Total Story Points Completed / Number of Sprints
Use last 3-5 sprints for rolling average. More data = more reliable forecast.
Sprints Needed to Complete Backlog
Sprints Needed = Remaining Story Points / Average Velocity (round UP)
Always round UP — you cannot complete a partial sprint.
Time to Release
Time = Sprints Needed x Sprint Length (weeks)
Kanban Throughput
Throughput = Items Completed / Time Period
Example: 24 user stories in 4 weeks = 6 stories/week throughput
📋 Sprint Capacity Math
Sprint Capacity (Hours)
Capacity = Team Members x Working Days x Hours/Day x Availability %
Availability % = typically 70-80% (accounts for meetings, admin, support, email)
Capacity in Story Points
Point Capacity = Hour Capacity / Avg Hours per Story Point
Sprint Capacity Example:
7 developers | 10 working days | 8 hrs/day | 75% availability
Capacity = 7 x 10 x 8 x 0.75 = 420 hours
If avg story = 7 hours: Point capacity = 420 / 7 = 60 story points/sprint
But team velocity (historical) = 48 pts. Plan based on velocity (48), not capacity (60).
7 developers | 10 working days | 8 hrs/day | 75% availability
Capacity = 7 x 10 x 8 x 0.75 = 420 hours
If avg story = 7 hours: Point capacity = 420 / 7 = 60 story points/sprint
But team velocity (historical) = 48 pts. Plan based on velocity (48), not capacity (60).
📉 Burndown & Burnup Charts
Ideal Daily Burndown Rate
Daily Rate = Total Sprint Points / Sprint Working Days
Creates the "ideal line" — straight diagonal from start to zero
Remaining Work at Any Day
Remaining = Starting Points - Points Completed So Far
Burnup Chart Progress
Scope line (flat/rising) vs Completed line (rising toward scope)
When scope line RISES = scope creep added. Burnup makes this visible; burndown does NOT.
📅 Release Planning Math
Features Deliverable by Fixed Date
Deliverable Points = Available Sprints x Average Velocity
If backlog > deliverable points: must defer lowest-priority items or negotiate deadline
Release Planning Scenario:
Release in 4 months. Sprint = 2 weeks. Available sprints = 8.
Team velocity = 35 pts/sprint. Deliverable = 8 x 35 = 280 story points
Total backlog = 380 points. Deficit = 100 points must be deferred.
PM works with Product Owner to prioritize top 280 points for this release.
Release in 4 months. Sprint = 2 weeks. Available sprints = 8.
Team velocity = 35 pts/sprint. Deliverable = 8 x 35 = 280 story points
Total backlog = 380 points. Deficit = 100 points must be deferred.
PM works with Product Owner to prioritize top 280 points for this release.
⭐ Agile Metrics Interpretation Table
| Metric | Value / Trend | Status | Meaning & Action |
|---|---|---|---|
| Velocity | Stable or rising | Healthy ✓ | Team improving or consistent — reliable for forecasting |
| Velocity | Declining or erratic | Problem ✗ | Team impediments, scope changes, or instability — investigate |
| Burndown | Below ideal line | Ahead of plan ✓ | Team completing more than expected — may finish early |
| Burndown | Above ideal line | Behind plan ✗ | Team falling behind — risk of not completing sprint goals |
| Burndown | Flat line (no descent) | STALLED ✗ | No work being completed — major impediment exists |
| Burnup Scope Line | Rising | Scope Creep ✗ | New work added — delivery date will be pushed out |
| Sprint Capacity vs Velocity | Capacity > Velocity | Normal gap | Overhead consumes 20-30% — this is expected and healthy |
| Sprints Needed | Decreasing each sprint | On track ✓ | Backlog being consumed as planned |
| Sprints Needed | Not decreasing | Problem ✗ | Scope added as fast as being completed — "treadmill effect" |
📘 Agile Math Scenarios — 5 Cases
Case 1 — How Many Sprints Remaining?
Sprints 1-4 velocity: 32, 28, 36, 30 pts. Remaining backlog: 195 pts.
Average velocity = (32+28+36+30)/4 = 126/4 = 31.5 pts/sprint
Sprints needed = 195/31.5 = 6.19 → 7 sprints (always round UP)
Sprint = 2 weeks → 14 weeks remaining
Average velocity = (32+28+36+30)/4 = 126/4 = 31.5 pts/sprint
Sprints needed = 195/31.5 = 6.19 → 7 sprints (always round UP)
Sprint = 2 weeks → 14 weeks remaining
Case 2 — Will We Make the Release Date?
Stakeholder wants release in 10 weeks. Sprint = 2 weeks = 5 sprints available.
Velocity = 40 pts/sprint. Deliverable = 5 x 40 = 200 points
Remaining backlog = 240 pts. Shortfall = 40 pts.
Options: Remove 40 pts from release scope, OR add one more sprint (extend 2 weeks), OR increase velocity.
Velocity = 40 pts/sprint. Deliverable = 5 x 40 = 200 points
Remaining backlog = 240 pts. Shortfall = 40 pts.
Options: Remove 40 pts from release scope, OR add one more sprint (extend 2 weeks), OR increase velocity.
Case 3 — Reading a Burndown Chart
Sprint: 60 pts, 10 days. Ideal rate = 6 pts/day.
Day 3 actual: 30 pts completed → Remaining = 30 pts (ideal at day 3 = 42 remaining)
30 remaining vs 42 ideal → Ahead of plan by 12 points GOOD
Day 7 actual: 44 pts completed → Remaining = 16 pts (ideal = 18 remaining)
16 vs 18 → Still slightly ahead GOOD
Day 3 actual: 30 pts completed → Remaining = 30 pts (ideal at day 3 = 42 remaining)
30 remaining vs 42 ideal → Ahead of plan by 12 points GOOD
Day 7 actual: 44 pts completed → Remaining = 16 pts (ideal = 18 remaining)
16 vs 18 → Still slightly ahead GOOD
Case 4 — Scope Creep on Burnup Chart
🎭 Burnup Reveals Hidden Scope Creep:
Sprint 1: Scope=100pts (line flat). Completed rises to 30.
Sprint 2: Scope jumps to 120pts (scope line rises!). Completed rises to 55.
Sprint 3: Scope at 120. Completed rises to 80.
PM notices: 20 new points added in Sprint 2. Burndown chart would NOT show this — scope looks "normal."
Burnup REVEALS the addition — prompts discussion with PO about source of new scope.
Sprint 1: Scope=100pts (line flat). Completed rises to 30.
Sprint 2: Scope jumps to 120pts (scope line rises!). Completed rises to 55.
Sprint 3: Scope at 120. Completed rises to 80.
PM notices: 20 new points added in Sprint 2. Burndown chart would NOT show this — scope looks "normal."
Burnup REVEALS the addition — prompts discussion with PO about source of new scope.
Case 5 — Agile EVM Hybrid
Sprint planned: 50 pts (PV=50). Completed: 42 pts (EV=42). Days used: 10 (AC=10 days).
SPI = 42/50 = 0.84 BEHIND (84% schedule efficiency)
If 300 pts remain, at current rate: 300/42 = 7.14 sprints → 8 sprints needed
At 0.84 efficiency, consider: address impediments OR reduce scope.
SPI = 42/50 = 0.84 BEHIND (84% schedule efficiency)
If 300 pts remain, at current rate: 300/42 = 7.14 sprints → 8 sprints needed
At 0.84 efficiency, consider: address impediments OR reduce scope.
🎲 Probability Basics
Basic Probability
P(Event) = Favorable Outcomes / Total Possible Outcomes
Range: 0 (impossible) to 1.0 (certain). Expressed as decimal or %.
Complement Rule
P(NOT A) = 1 - P(A)
"Probability event does NOT happen" = 1 minus probability it DOES happen
AND Rule — Both events must happen (independent)
P(A AND B) = P(A) x P(B)
For independent events only. Example: P(no delay in Phase 1 AND Phase 2)
OR Rule — Either event occurs
P(A OR B) = P(A) + P(B) - P(A AND B)
For mutually exclusive events (can't both happen): P(A OR B) = P(A) + P(B)
⭐ Probability Interpretation Table
| Probability Value | Qualitative Level | Meaning | Risk Response Priority |
|---|---|---|---|
| 0.90 - 1.00 (90-100%) | Very High | Near certainty — treat as fact for planning | Highest — must respond |
| 0.70 - 0.89 (70-89%) | High | Likely to occur — significant probability | High priority response needed |
| 0.30 - 0.69 (30-69%) | Medium | Could go either way — monitor closely | Moderate response appropriate |
| 0.10 - 0.29 (10-29%) | Low | Unlikely but possible — watchlist | Accept or low-cost mitigation |
| 0.01 - 0.09 (1-9%) | Very Low | Rare occurrence — mostly accept | Accept; note in risk register |
| Sequential Success Probability (AND Rule) | |||
| 3 activities x 90% each | 0.9 x 0.9 x 0.9 = 0.729 = 72.9% overall success. Even 90% per step drops to 73% over 3 steps! | ||
| 5 activities x 80% each | 0.8^5 = 0.328 = 32.8% overall. Risk accumulates rapidly in serial activities. | ||
Scenario — Project Success Through 4 Gates:
Each gate approval probability: 90%, 85%, 95%, 80%
P(all approved) = 0.90 x 0.85 x 0.95 x 0.80 = 0.582 = 58.2%
P(at least one rejected) = 1 - 0.582 = 41.8% risk of project stoppage
Even with high individual probabilities, overall risk is significant in sequential processes.
Each gate approval probability: 90%, 85%, 95%, 80%
P(all approved) = 0.90 x 0.85 x 0.95 x 0.80 = 0.582 = 58.2%
P(at least one rejected) = 1 - 0.582 = 41.8% risk of project stoppage
Even with high individual probabilities, overall risk is significant in sequential processes.
🎲 Monte Carlo Simulation
Monte CarloMonte Carlo runs thousands of project simulations using probability distributions for each variable. It outputs a probability distribution of possible project completion dates or costs, typically shown as an S-curve or histogram with percentile values. — runs 10,000+ simulations to model project uncertainty.
Reading Monte Carlo Output — Confidence Levels
P80 = value with 80% probability project finishes at or below this
P50 = 50% chance | P80 = recommended | P90 = conservative | P10 = very optimistic
| Percentile | Meaning | Risk Level | Use Case |
|---|---|---|---|
| P10 | 10% chance of finishing at or below this cost/date | Very risky — optimistic | Best-case scenario only |
| P50 | 50% chance of finishing at or below (median) | Moderate risk | Most likely scenario — not recommended for funding |
| P80 | 80% chance of finishing at or below | Low-moderate risk | Recommended for budget requests and commitments |
| P90 | 90% chance of finishing at or below | Very low risk | High-risk projects, critical deliverables |
Monte Carlo Output Example:
Simulation results for project completion cost:
P10 = $880,000 (90% chance of costing MORE than this)
P50 = $1,050,000 (50/50 — median outcome)
P80 = $1,190,000 Recommended budget request
P90 = $1,310,000 (conservative — used for critical infrastructure)
If PM requests $1,050,000 (P50), there is a 50% chance of overrun. Request P80 = $1,190,000 for responsible budgeting.
Simulation results for project completion cost:
P10 = $880,000 (90% chance of costing MORE than this)
P50 = $1,050,000 (50/50 — median outcome)
P80 = $1,190,000 Recommended budget request
P90 = $1,310,000 (conservative — used for critical infrastructure)
If PM requests $1,050,000 (P50), there is a 50% chance of overrun. Request P80 = $1,190,000 for responsible budgeting.
💰 Reserve Analysis
Contingency Reserve
Contingency = Sum of (Probability x Impact) for all KNOWN risks
For "known unknowns." PM has full authority to use. Part of cost baseline.
Management Reserve
Management Reserve = % of Project Cost (typically 5-15%)
For "unknown unknowns" — unforeseen events. Requires management approval. NOT in cost baseline.
⭐ Reserve Interpretation Table
| Reserve Type | For What Risks? | Who Approves Use? | In Cost Baseline? | In Project Budget? |
|---|---|---|---|---|
| Contingency Reserve | Known unknowns (identified risks) | Project Manager ✓ | YES | YES |
| Management Reserve | Unknown unknowns (unforeseen) | Management / Sponsor | NO | YES |
| Budget Hierarchy | ||||
|
Planned Cost + Contingency Reserve = Cost Baseline Cost Baseline + Management Reserve = Project Budget Example: Planned=$500K + Contingency=$40K = Baseline $540K | + Mgmt Reserve $54K = Budget $594K | ||||
🎭 Exam Trap — Which reserve does PM need approval for?
"The project encountered an unforeseen geological issue. PM needs $30K." — This is an unknown unknown = Management Reserve = needs management approval.
"Risk R4 occurred (was in risk register). PM uses $15K reserve." — This is a known risk = Contingency Reserve = PM authorized to use independently.
"The project encountered an unforeseen geological issue. PM needs $30K." — This is an unknown unknown = Management Reserve = needs management approval.
"Risk R4 occurred (was in risk register). PM uses $15K reserve." — This is a known risk = Contingency Reserve = PM authorized to use independently.
👥 Resource Loading & Utilization
Resource Utilization %
Utilization = (Assigned Hours / Available Hours) x 100%
>100% = over-allocated (problem) | 80-90% = practical sweet spot | 100% = no buffer
Effort vs Duration
Duration = Effort (person-days) / Number of People
2 people x 5 days = 10 person-days effort. Adding people shortens duration but adds communication.
⭐ Resource Utilization Interpretation Table
| Utilization % | Status | Meaning | Action |
|---|---|---|---|
| < 60% | Under-utilized | Resource has excess capacity — could take more work | Assign additional tasks or reassign |
| 70-85% | Optimal ✓ | Productive with buffer for issues and admin | Maintain — this is the target range |
| 85-100% | Watch closely | High but sustainable short-term; no room for surprises | Monitor; avoid adding more work |
| > 100% | OVER-ALLOCATED ✗ | Resource physically cannot do all assigned work | Resolve: reassign, delay, or add resources |
| Leveling vs Smoothing | |||
| Resource Leveling | May extend schedule | Delays activities to resolve over-allocation. Schedule floats out. | |
| Resource Smoothing | Schedule preserved | Uses float to adjust assignments. Does NOT extend project end date. | |
Resource Calculation Scenario:
Inspector A: Available 40 hrs/week. Assigned: Project Alpha=22hrs + Project Beta=20hrs + Project Gamma=8hrs
Total assigned = 50 hrs. Utilization = 50/40 = 125% — OVER-ALLOCATED
Resolution options: Reduce hours on one project, extend timeline, or bring in second inspector.
Inspector A: Available 40 hrs/week. Assigned: Project Alpha=22hrs + Project Beta=20hrs + Project Gamma=8hrs
Total assigned = 50 hrs. Utilization = 50/40 = 125% — OVER-ALLOCATED
Resolution options: Reduce hours on one project, extend timeline, or bring in second inspector.
🎭 Brooks's Law — Adding People Makes It Worse (initially):
Team of 8: Channels = 8x7/2 = 28 | Add 4 people late in project: 12 people = 12x11/2 = 66 channels
Added 4 people but added 38 new communication channels. Training time + onboarding can DELAY completion.
The exam expects you to know: adding people to a LATE project usually makes it LATER.
Team of 8: Channels = 8x7/2 = 28 | Add 4 people late in project: 12 people = 12x11/2 = 66 channels
Added 4 people but added 38 new communication channels. Training time + onboarding can DELAY completion.
The exam expects you to know: adding people to a LATE project usually makes it LATER.
📈 Learning Curve
Learning Curve Effect
New Unit Time = Previous Doubling Time x Learning Rate
Applied each time cumulative output DOUBLES. 80% curve: each doubling = 80% of previous time.
| Learning Rate | Effect | Industry |
|---|---|---|
| 70% | Very rapid learning — strong improvement | Complex manufacturing, aerospace |
| 80% | Good learning rate — typical | Construction, software, engineering |
| 90% | Slow learning — minimal improvement | Mature processes, repetitive tasks |
| 100% | No learning — constant time per unit | Fully automated processes |
80% Learning Curve Example — Welding:
Unit 1: 100 hrs | Unit 2 (1st doubling): 100x0.80 = 80 hrs | Unit 4 (2nd doubling): 80x0.80 = 64 hrs | Unit 8: 64x0.80 = 51.2 hrs
Cost reduction over time: significant savings in repetitive construction work (bridge sections, drainage structures).
Unit 1: 100 hrs | Unit 2 (1st doubling): 100x0.80 = 80 hrs | Unit 4 (2nd doubling): 80x0.80 = 64 hrs | Unit 8: 64x0.80 = 51.2 hrs
Cost reduction over time: significant savings in repetitive construction work (bridge sections, drainage structures).
🔧 Make-or-Buy Analysis
Break-Even Units (Make vs Buy)
Break-Even = Fixed Cost of Making / (Buy Price per Unit - Variable Cost of Making)
Below break-even: BUY cheaper. Above break-even: MAKE cheaper.
Total Make Cost
Make Cost = Fixed Setup Cost + (Variable Cost x Units)
Total Buy Cost
Buy Cost = Buy Price x Units
⭐ Make-or-Buy Interpretation Table
| Scenario | Decision | Reason |
|---|---|---|
| Quantity < Break-Even Units | BUY ✓ | Fixed setup cost not recovered — buying is cheaper per unit |
| Quantity = Break-Even Units | EQUAL | Total cost is exactly the same — decide on other factors |
| Quantity > Break-Even Units | MAKE ✓ | High volume recovers setup cost — making is cheaper overall |
| Other Factors Beyond Cost | ||
| Core competency? | MAKE | Keeps critical skills in-house, protects IP |
| Specialized expertise needed? | BUY | Vendor has specialized capability unavailable in-house |
| Capacity constraints? | BUY | Internal team cannot take on additional work |
| Long-term strategic value? | MAKE | Build internal capability for future benefit |
📘 Procurement Scenarios — 3 Cases
Case 1 — Make-or-Buy Calculation:
Buy price from vendor: $80/unit. In-house: Setup=$30,000 + $50/unit variable cost.
Break-even = $30,000 / ($80-$50) = $30,000/$30 = 1,000 units
If need 800 units: Buy (800x$80=$64K) vs Make ($30K+800x$50=$70K) → BUY saves $6,000
If need 1,500 units: Buy (1500x$80=$120K) vs Make ($30K+1500x$50=$105K) → MAKE saves $15,000
Buy price from vendor: $80/unit. In-house: Setup=$30,000 + $50/unit variable cost.
Break-even = $30,000 / ($80-$50) = $30,000/$30 = 1,000 units
If need 800 units: Buy (800x$80=$64K) vs Make ($30K+800x$50=$70K) → BUY saves $6,000
If need 1,500 units: Buy (1500x$80=$120K) vs Make ($30K+1500x$50=$105K) → MAKE saves $15,000
Case 2 — Which Contract Type for Uncertain Scope?
Situation: IT system requirements still evolving. Duration uncertain. Expertise needed is rare.
Options: FFP (fixed price) — risky for seller → seller adds huge contingency to price.
Better choice: T&M with Not-to-Exceed cap — buyer pays for actual time, seller not penalized for scope changes. Cap protects buyer.
OR: CPIF — cost reimbursable with incentive to control cost.
Situation: IT system requirements still evolving. Duration uncertain. Expertise needed is rare.
Options: FFP (fixed price) — risky for seller → seller adds huge contingency to price.
Better choice: T&M with Not-to-Exceed cap — buyer pays for actual time, seller not penalized for scope changes. Cap protects buyer.
OR: CPIF — cost reimbursable with incentive to control cost.
🎭 Exam Trap — Which contract has highest buyer risk?
"Which contract type puts the most risk on the BUYER?" → CPFF (Cost Plus Fixed Fee)
Seller gets same fee regardless of performance. Buyer pays ALL cost overruns.
"Which puts most risk on SELLER?" → FFP (Firm Fixed Price)
Seller must deliver for fixed price — any overrun comes from seller profit.
"Which contract type puts the most risk on the BUYER?" → CPFF (Cost Plus Fixed Fee)
Seller gets same fee regardless of performance. Buyer pays ALL cost overruns.
"Which puts most risk on SELLER?" → FFP (Firm Fixed Price)
Seller must deliver for fixed price — any overrun comes from seller profit.
💹 Break-Even Analysis
Break-Even Units
Break-Even = Fixed Costs / (Selling Price - Variable Cost per Unit)
(Selling Price - Variable Cost) = Contribution Margin per Unit
Break-Even Revenue
Break-Even Revenue = Fixed Costs / Contribution Margin Ratio
Contribution Margin Ratio = (Price - Variable Cost) / Price
Profit at Given Sales Volume
Profit = (Units - Break-Even Units) x Contribution Margin per Unit
⭐ Break-Even Interpretation Table
| Sales Level | Result | Meaning |
|---|---|---|
| Sales < Break-Even | LOSS ✗ | Fixed costs not fully recovered — operating at a loss |
| Sales = Break-Even | ZERO profit/loss | All costs covered, no profit yet — neutral |
| Sales > Break-Even | PROFIT ✓ | Every unit above break-even generates pure profit (contribution margin) |
| Sensitivity Analysis | ||
| Fixed costs INCREASE | Break-even RISES | Must sell more to cover higher overhead |
| Selling price INCREASES | Break-even FALLS | Each unit contributes more — fewer needed to cover fixed costs |
| Variable cost INCREASES | Break-even RISES | Less profit per unit — need more sales to break even |
📘 Financial Math Scenarios — 4 Cases
Case 1 — Break-Even for New Service
Monthly fixed costs: $12,000. Service price: $150. Variable cost per client: $60.
Contribution margin = $150 - $60 = $90 per client.
Break-even = $12,000 / $90 = 133.3 → 134 clients/month
At 200 clients: Profit = (200-134) x $90 = 66 x $90 = $5,940/month
Contribution margin = $150 - $60 = $90 per client.
Break-even = $12,000 / $90 = 133.3 → 134 clients/month
At 200 clients: Profit = (200-134) x $90 = 66 x $90 = $5,940/month
Case 2 — PV and FV for Investment Decision
Invest $50,000 today OR receive $70,000 in 5 years. Discount rate = 8%.
PV of $70,000 = $70,000 / (1.08)^5 = $70,000 / 1.469 = $47,651
PV ($47,651) < Cost ($50,000) → Do NOT invest — not worth it at 8% discount rate.
PV of $70,000 = $70,000 / (1.08)^5 = $70,000 / 1.469 = $47,651
PV ($47,651) < Cost ($50,000) → Do NOT invest — not worth it at 8% discount rate.
Case 3 — NPV Project Selection
Discount rate = 12%. Project Alpha requires $150,000 investment.
Year 1: $60,000 → PV = $60K/1.12 = $53,571
Year 2: $65,000 → PV = $65K/1.2544 = $51,818
Year 3: $70,000 → PV = $70K/1.4049 = $49,823
Total PV = $155,212 - $150,000 = NPV = +$5,212 → ACCEPT Project Alpha
Project Beta: NPV = +$8,400 → If mutually exclusive, choose Beta (higher NPV)
Year 1: $60,000 → PV = $60K/1.12 = $53,571
Year 2: $65,000 → PV = $65K/1.2544 = $51,818
Year 3: $70,000 → PV = $70K/1.4049 = $49,823
Total PV = $155,212 - $150,000 = NPV = +$5,212 → ACCEPT Project Alpha
Project Beta: NPV = +$8,400 → If mutually exclusive, choose Beta (higher NPV)
Case 4 — Sunk Cost vs Future Value Decision
Engineering firm has spent $800K on a highway design project. New analysis shows:
- Cost to complete: $400K more
- Expected contract value if completed: $1.3M
- Penalty for stopping: $50K
Sunk cost = $800K → IGNORE.
Continue: spend $400K, receive $1.3M → Net future gain = +$900K
Stop: pay $50K penalty → Net future = -$50K
Decision: CONTINUE (future gain of $900K vs $50K loss)
The $800K already spent is irrelevant — only future flows matter.
- Cost to complete: $400K more
- Expected contract value if completed: $1.3M
- Penalty for stopping: $50K
Sunk cost = $800K → IGNORE.
Continue: spend $400K, receive $1.3M → Net future gain = +$900K
Stop: pay $50K penalty → Net future = -$50K
Decision: CONTINUE (future gain of $900K vs $50K loss)
The $800K already spent is irrelevant — only future flows matter.
📊 Estimate Type Accuracy Table
| Estimate Type | Accuracy Range | When Used | Basis | Cost/Effort |
|---|---|---|---|---|
| Order of Magnitude (ROM) | -25% to +75% | Project initiation — very early | Expert judgment, past projects | Low |
| Budget Estimate | -10% to +25% | Early planning | Analogous, parametric | Moderate |
| Definitive Estimate | -5% to +10% | Detailed planning | Bottom-up, detailed WBS | High |
Applying Estimate Ranges:
PM gives ROM estimate of $500,000 at project initiation.
Expected range: $500K x (1-0.25) to $500K x (1+0.75) = $375,000 to $875,000
Sponsor should not commit to exact budget until a definitive estimate is available.
PM gives ROM estimate of $500,000 at project initiation.
Expected range: $500K x (1-0.25) to $500K x (1+0.75) = $375,000 to $875,000
Sponsor should not commit to exact budget until a definitive estimate is available.
📂 WBS & Bottom-Up Estimating
Bottom-Up Total Estimate
Project Total = Sum of All Work Package Estimates
Most accurate method. Requires complete WBS. Time-consuming but reliable.
Work Package Size Rule (80-Hour Rule)
Work Package = 8 hours (min) to 80 hours (max)
Should be completable in 1-2 reporting periods. Too large = can't track. Too small = overhead.
Bottom-Up Estimate Scenario — Road Project:
WP1 (Clearing): $45,000 | WP2 (Grading): $120,000 | WP3 (Sub-base): $80,000
WP4 (Paving): $190,000 | WP5 (Striping): $25,000 | WP6 (Signage): $18,000
Project Total = $45K+$120K+$80K+$190K+$25K+$18K = $478,000 (definitive estimate, -5%/+10%)
Low range: $478K x 0.95 = $454,100 | High range: $478K x 1.10 = $525,800
WP1 (Clearing): $45,000 | WP2 (Grading): $120,000 | WP3 (Sub-base): $80,000
WP4 (Paving): $190,000 | WP5 (Striping): $25,000 | WP6 (Signage): $18,000
Project Total = $45K+$120K+$80K+$190K+$25K+$18K = $478,000 (definitive estimate, -5%/+10%)
Low range: $478K x 0.95 = $454,100 | High range: $478K x 1.10 = $525,800
📝 Master Cheat Sheet — Part 2 Formulas
SCHEDULE COMPRESSION
- Cost Slope = (Crash Cost-Normal Cost)/(Normal-Crash Duration)
- Crash LOWEST cost slope FIRST on critical path only
- Multiple critical paths: must crash BOTH simultaneously
- Fast-track: overlap activities = no direct cost, HIGH risk
AGILE MATH
- Velocity = Points Completed / Sprints (rolling average)
- Sprints Needed = Remaining Points / Velocity (round UP)
- Sprint Capacity = Members x Days x Hours x Availability%
- Burndown: below ideal line = GOOD | Flat = stalled
- Burnup: scope line rising = SCOPE CREEP
- Deliverable by date = Available Sprints x Velocity
PROBABILITY / RISK / RESERVES
- P(NOT A) = 1 - P(A)
- P(A AND B) = P(A) x P(B) [independent]
- Monte Carlo P80 = recommended confidence level for budgets
- Contingency = sum of EMVs [PM uses | in baseline]
- Management Reserve = % of budget [management uses | NOT in baseline]
- Cost Baseline = Planned + Contingency | Budget = Baseline + Mgmt Reserve
RESOURCE / MAKE-BUY / FINANCIAL
- Utilization = Assigned/Available x 100% [>100% = problem]
- Break-even units = Fixed Cost / (Buy Price - Variable Cost)
- Break-even for new product = Fixed / Contribution Margin per unit
- ROM: -25%/+75% | Budget: -10%/+25% | Definitive: -5%/+10%
- PV = FV/(1+r)^n | FV = PV x (1+r)^n | NPV>0 = accept
🎯 Advanced PMP Math Exam Tips — Part 2
- Crash lowest cost slope first, critical path ONLY. Non-critical crashing = wasted money, zero time saved.
- When two critical paths exist: must crash BOTH simultaneously. Add both cost slopes for each day saved.
- Fast-track = more risk, less cost. Crash = more cost, less risk. Know the trade-off cold.
- Velocity is historical fact. Capacity is theoretical max. Always plan releases using VELOCITY.
- Always round sprints UP — you cannot have 7.3 sprints. Round 7.3 to 8 sprints.
- Burnup shows scope creep; burndown does NOT. Burndown only shows remaining work — scope additions are invisible.
- Monte Carlo P80 = recommended for budget/schedule commitments. P50 is too risky (50% overrun probability).
- Contingency = PM controls. Management Reserve = needs approval. Cost baseline includes contingency; project budget includes both.
- Sequential risk compounds rapidly. Three 90% success rates = 72.9% overall — risk accumulates in serial processes.
- Utilization >100% = over-allocated. Optimal range is 70-85% (room for admin, meetings, issues).
- Resource leveling MAY extend schedule. Resource smoothing uses existing float and preserves end date.
- Learning curve: each DOUBLING of output (not each unit) triggers the reduction. 80% curve = 80% of previous doubling time.
- Make-or-Buy: calculate total cost at GIVEN quantity, not just unit cost. Setup cost changes everything.
- CPFF has HIGHEST buyer risk. FFP has HIGHEST seller risk. Know this for contract selection questions.
- Sunk cost = IGNORE for future decisions. Only future costs and benefits matter in continuation decisions.