Because in real projects, schedule risk and cost risk are rarely independent. If a key work package takes longer, labor costs rise. If a supplier slips, expediting costs appear. If testing runs long, both the finish date and the burn rate move in the wrong direction together. Looking at time risk and cost risk separately can create false confidence.<\/p>\n
A better approach is to model them together and ask a more useful question: How confident are we that the project will finish by this date and under this cost at the same time?<\/strong> That is the idea behind an integrated cost and schedule risk model. And once you visualize it as a surface with time, cost, and confidence<\/strong>, you get a practical 3D view that leaders can actually use.<\/p>\n If you tell a sponsor, \u201cWe have an 80% confidence in the schedule\u201d and then say, \u201cWe also have an 80% confidence in the budget,\u201d it sounds reassuring. But that does not<\/strong> automatically mean you have an 80% confidence of achieving both together.<\/p>\n That matters because projects are judged on combined performance. Finishing on time but far over budget is not a win. Finishing under budget but months late usually is not either.<\/p>\n Here is the plain-language issue:<\/p>\n Even in a simplified world where time and cost were fully unrelated, 80% confidence on each side would only translate to 64% confidence of getting both. In real projects, they are usually linked, which is exactly why the combined picture matters more than the separate ones.<\/p>\n This is not about making project controls more academic. It is about making the answer match the real decision.<\/p>\n Joint confidence is simply the probability that two things are true at the same time<\/strong>:<\/p>\n If you are used to percentile language, you can think of it like this:<\/p>\n That is why leaders care. A portfolio review, steering committee, or capital approval board is not really asking for two separate promises. It is asking for one practical outcome: How likely is this project to land successfully within the approved box?<\/em><\/p>\n Imagine a simple target box:<\/p>\n A joint confidence calculation answers: What percentage of simulated project outcomes fall inside that box?<\/em><\/p>\n Once you frame the problem that way, the logic becomes much clearer.<\/p>\n The term \u201c3D confidence model\u201d sounds more technical than it needs to be.<\/p>\n Think of it as three dimensions:<\/p>\n In practice, that means you build many possible project<\/p>\n outcomes through simulation, then map how often each combination of duration<\/strong> and cost<\/strong> occurs.<\/p>\n You can picture it in a few different ways:<\/p>\n The point is not the graphics by themselves. The point is that a single view can show:<\/p>\n That is far more useful than placing one schedule S-curve on one slide and one cost S-curve on another and hoping people mentally combine them correctly.<\/p>\n At a high level, an integrated cost and schedule risk model combines three things:<\/p>\n Then it runs repeated simulations to produce a cloud of possible outcomes.<\/p>\n The schedule needs more than task dates in a spreadsheet. It should reflect actual delivery logic:<\/p>\n For risk modeling, each key activity or work package typically gets an uncertainty range, such as:<\/p>\n Or, if the data is mature enough, a fitted probability distribution.<\/p>\n The cost model should be connected to how the project actually spends money. Typical categories include:<\/p>\n Some costs behave mostly as fixed amounts. Others are strongly time-dependent.<\/p>\n For example:<\/p>\n This is where integration becomes important. If cost is modeled as though time changes do not matter, the model misses the way projects really behave.<\/p>\n This is the part that separates a realistic model from a decorative one.<\/p>\n Uncertainty means key inputs can vary. Correlation means some of those variations move together.<\/p>\n Examples:<\/p>\n Without correlation, a simulation may underestimate how often bad outcomes cluster together.<\/p>\n A typical Monte Carlo simulation runs thousands of trials.<\/p>\n In each trial, the model does things like:<\/p>\n The result from one trial is a pair:<\/p>\n After thousands of trials, you have a large set of possible project outcomes. That outcome set can then be plotted as:<\/p>\n That surface is just a way of summarizing the same simulation evidence in a form that supports decisions.<\/p>\n A good risk model should make decisions easier, not harder.<\/p>\n Here are the main questions leaders can ask when looking at an integrated confidence view.<\/p>\n This shows the combinations of duration and cost that occur most often in the simulation.<\/p>\n It is often more informative than a single deterministic baseline because it reveals the likely neighborhood, not just one point estimate.<\/p>\n Suppose the approved target is:<\/p>\n The model can show whether that point sits in:<\/p>\n If it sits in a low-confidence area, the project may be carrying more commitment risk than leadership realizes.<\/p>\n In some projects, a small gain in schedule confidence may require a large increase in expected cost. In others, modest cost relief may create a large increase in schedule exposure.<\/p>\n The shape of the surface helps reveal whether tradeoffs are gentle or severe.<\/p>\n If the outcome cloud stretches diagonally, that often indicates a strong relationship between delay and cost growth.<\/p>\n If it is more circular or diffuse, the link may be weaker.<\/p>\n This matters when choosing mitigation actions. A strongly linked project may benefit from interventions that reduce root-cause uncertainty rather than treating time and cost separately.<\/p>\n Assume a project simulation produces the following separate results:<\/p>\n A sponsor might be tempted to think, \u201cFine, let us approve Day 125 and $950,000.\u201d<\/p>\n But the integrated model may show something like this:<\/p>\n That changes the conversation.<\/p>\n If the sponsor wants a joint confidence<\/strong> of 80%, the target box may need to move to something like:<\/p>\n The exact numbers depend on the modeled relationship between time and cost, but the principle stays the same: separate P-values do not answer the combined question.<\/p>\n Executive decisions are usually binary in practice:<\/p>\n Those decisions should be based on the confidence of the actual commitment<\/strong>, not on two disconnected confidence statements.<\/p>\n An integrated model helps governance in at least five ways.<\/p>\n Approvals can be tied to the confidence of the combined target, not just optimistic baseline numbers.<\/p>\n Contingency can be discussed in relation to both time and cost together. This is often more practical than holding a cost contingency that assumes no schedule movement.<\/p>\n If the approved box corresponds to low joint confidence, that can trigger stronger review, phased authorization, or added mitigation before commitment.<\/p>\n Two projects may each claim \u201cP80\u201d status, but one may have much lower joint confidence once cost and schedule are integrated. That gives portfolio boards a more consistent basis for comparison.<\/p>\n Leaders can say, in plain terms, \u201cAt the approved date and budget together, we currently have a 62% confidence,\u201d which is far clearer than presenting two separate charts and leaving the audience to infer the rest.<\/p>\n If there is one concept that deserves extra attention, it is correlation.<\/p>\n Many project models include uncertainty, but not enough meaningful dependency between variables. That can create a misleadingly smooth picture.<\/p>\n Correlation appears when one underlying cause affects multiple outcomes at once.<\/p>\n Examples include:<\/p>\n If the model assumes that schedule and cost move mostly independently, it may generate too many mixed outcomes like:<\/p>\n Some projects do produce those patterns, but many do not. In many delivery environments, bad schedule outcomes and bad cost outcomes tend to arrive together.<\/p>\n When that is true, a weakly correlated model can overstate joint confidence.<\/p>\n Do not add correlation everywhere just to appear sophisticated. Add it where there is a credible causal connection and where the effect is material.<\/p>\n A few strong, well-justified dependencies are usually better than dozens of arbitrary correlations.<\/p>\n Teams often hesitate because they think integrated modeling requires perfect data. It does not.<\/p>\n What it requires is structured judgment plus traceable assumptions<\/strong>.<\/p>\n You can build a useful model if you have:<\/p>\n Of course, better data improves the model. But the absence of perfection is not a reason to keep making commitments from disconnected views.<\/p>\n A simple integrated model with explicit assumptions is usually better than two polished but unrelated forecasts.<\/p>\n Integrated models can be very powerful, but only if they are built carefully.<\/p>\n If the logic is weak, simulation quality will also be weak. The model must reflect how work really flows.<\/p>\n Time-dependent cost needs to react to time movement.<\/p>\n If a risk is already represented through uncertain activity durations and also added again as a separate cost event without adjustment, the model can overstate exposure.<\/p>\n A model should reflect current mitigation and response assumptions. Otherwise it may estimate a risk profile that no longer matches the actual delivery strategy.<\/p>\n A 3D chart can look impressive, but visual polish does not guarantee validity. The assumptions and logic still matter more than the graphics.<\/p>\n One of the best ways to present integrated confidence is to avoid leading with math.<\/p>\n Instead, use a sequence like this:<\/p>\n For example:<\/p>\n Our current approved target is June 30 and $12.5M. Based on the integrated simulation, 58% of outcomes achieve both together. The main reasons outcomes miss the box are vendor package delay, late design release, and field productivity uncertainty. If we lock the vendor earlier and reduce design interface uncertainty, the joint confidence improves materially.<\/p>\n<\/blockquote>\n That is a decision-ready message.<\/p>\n A mature integrated model can produce more than just one headline figure.<\/p>\n You can test many combinations of date and budget, not just the currently approved one.<\/p>\n Instead of asking, \u201cWhat confidence do we have at this target?\u201d you can ask, \u201cWhat target pair gives us 70% or 80% joint confidence?\u201d<\/p>\n You can also ask practical questions such as:<\/p>\n You can identify which uncertainty sources do the most to reduce joint confidence. That helps focus mitigation money where it matters.<\/p>\n The value of integrated risk modeling is not the model itself. It is what management does because of it.<\/p>\n Typical actions include:<\/p>\n In other words, the model should shape decisions, not just reporting.<\/p>\n If your organization is new to integrated cost and schedule risk analysis, you do not need to start with a massive enterprise model.<\/p>\n A practical first version can follow these steps:<\/p>\n Model the major work packages and major risk drivers. Do not try to simulate every minor task.<\/p>\n Separate:<\/p>\n Use historical data where available, and expert elicitation where needed.<\/p>\n Document the few relationships that matter most, such as:<\/p>\n Start simple. A small set of credible dependencies is enough to make the model meaningfully more realistic.<\/p>\n Use enough trials to make the confidence estimates reasonably stable. The exact number depends on model complexity, but the principle is straightforward:<\/p>\n Do not stop at one answer. Compare:<\/p>\n That gives leadership options, not just a warning.<\/p>\n The first lightweight model should support specific actions, such as:<\/p>\n That is enough to create value without overengineering the first effort.<\/p>\n An integrated model is most useful when it is treated as a living decision tool, not as a one-time study.<\/p>\n Typical update points include:<\/p>\n The purpose of updating is not to produce endless analysis. It is to keep the commitment view aligned with current reality.<\/p>\n The best integrated models are not built by one specialist working alone.<\/p>\n Useful participation usually includes:<\/p>\n Why this matters:<\/p>\n Cross-functional input improves assumptions and increases trust in the result.<\/p>\n Once an integrated model exists, the meeting itself gets better.<\/p>\n Instead of asking:<\/p>\n the conversation becomes:<\/p>\n That is a much stronger governance discussion because it is tied directly to the actual decision.<\/p>\n A few simple habits can make the outputs easier to use.<\/p>\n The difference between 67% and 69% joint confidence is rarely the main issue. The important question is whether the commitment is:<\/p>\n Use the model to understand the risk position, not to pretend that every decimal place is meaningful.<\/p>\n Integrated models are especially valuable when comparing choices:<\/p>\n Often the clearest value is not the absolute confidence number, but how much a decision changes it.<\/p>\n If outcomes miss the target box, ask why. Usually a small number of causes explain a large share of the misses.<\/p>\n That is where management attention should go.<\/p>\n Not every decision needs the same confidence level.<\/p>\n For example:<\/p>\n The right level depends on the consequence of being wrong.<\/p>\n Integrated modeling is powerful, but it is not magic.<\/p>\n It still depends on:<\/p>\n It also cannot fully predict:<\/p>\n That does not reduce its value. It simply means the model should inform judgment, not replace it.<\/p>\n Organizations do not need to become experts overnight. A practical maturity path often looks like this:<\/p>\n This is where many teams begin. It is better than deterministic planning alone, but still leaves the combined commitment question unresolved.<\/p>\n At this stage, the team can estimate joint confidence for a chosen date and budget pair.<\/p>\n The organization starts modeling a limited set of meaningful correlations and time-driven cost behavior.<\/p>\n The model is used to compare mitigation options, procurement strategies, and execution plans.<\/p>\n Joint confidence becomes part of routine approvals, portfolio reviews, and commitment decisions.<\/p>\n Even moving from Level 1 to Level 2 can materially improve decision quality.<\/p>\n If you need to explain the whole idea in one minute, this usually works:<\/p>\n Separate schedule confidence and cost confidence do not tell us the confidence of achieving both together. Because time and cost often move together, we need an integrated model. By simulating duration and cost at the same time, we can estimate the probability of finishing by a chosen date and under a chosen budget simultaneously. That gives leadership a more honest basis for approvals, contingency, and mitigation decisions.<\/p>\n<\/blockquote>\n Projects are not delivered in two separate worlds, one for dates and one for dollars. They are delivered in one world where time and cost interact constantly.<\/p>\n That is why the real management question is not:<\/p>\n It is:<\/p>\n A 3D integrated cost and schedule confidence model helps answer that question directly. And when leaders can see time, cost, and confidence together<\/strong>, the conversation shifts from isolated forecasts to realistic commitments.<\/p>\n\n\n <\/p>\n\n\n\n Your current target is too aggressive<\/strong>, and the model says the chance of meeting both<\/strong> the date and the budget together is low.<\/p>\n\n\n\n Here is how to read each chart.<\/p>\n\n\n\n This is the most important chart.<\/p>\n\n\n\n What it shows:<\/p>\n\n\n\n Plain-English meaning:<\/p>\n\n\n\n This is the same story, but cleaner.<\/p>\n\n\n\n What it shows:<\/p>\n\n\n\n Plain-English meaning:<\/p>\n\n\n\n This chart is supposed to show:<\/p>\n\n\n\n So, for example:<\/p>\n\n\n\n But your output looks visually wrong or at least misleading.<\/p>\n\n\n\n Why:<\/p>\n\n\n\n So the correct interpretation of the concept is:<\/p>\n\n\n\n But this particular heatmap is not trustworthy for precise interpretation<\/strong> in its current form.<\/p>\n\n\n\n Your model is telling a very clear story:<\/p>\n\n\n\n You could summarize the result like this:<\/p>\n\n\n\n The simulation suggests our current target of 60 days and $260,000 is aggressive. Most modeled outcomes fall beyond one or both limits, and time and cost move together. This means delays are likely to drive additional cost, not just schedule slippage. To improve confidence, we would need either more contingency, a relaxed target, or actions that materially reduce the main risk drivers.<\/p>\n<\/blockquote>\n\n\n\n Use the scatter plot and contour plot as your primary visuals here. They are both telling a consistent story:<\/p>\n\n\n\n The heatmap needs fixing before you rely on it.<\/p>\n\n\n\n <\/p>\n","protected":false},"excerpt":{"rendered":" Most project leaders hear two different risk stories. One story is about time: Are we likely to finish by the target date? The other is about money: Are we likely to stay inside the approved budget? Those conversations often happen in separate meetings, with separate charts, and sometimes with separate […]<\/p>\n","protected":false},"author":1,"featured_media":1983,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[177],"tags":[189,203,91,187,56],"class_list":["post-2004","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-monte-carlo-risk-analysis","tag-cost-management","tag-deep-dive","tag-evm","tag-monte-carlo","tag-risk-management"],"yoast_head":"\nWhy separate risk views often mislead<\/h2>\n
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What \u201cjoint confidence\u201d means in plain language<\/h2>\n
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What a 3D confidence model actually shows<\/h2>\n
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How the model is built<\/h2>\n
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1. Start with the schedule logic<\/h3>\n
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2. Link cost to the work<\/h3>\n
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3. Add uncertainty and correlation<\/h3>\n
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What gets simulated<\/h2>\n
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Reading the surface without getting lost in statistics<\/h2>\n
1. Where is the highest-density region?<\/h3>\n
2. Is the approved target inside a strong-confidence area?<\/h3>\n
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3. How steep is the tradeoff?<\/h3>\n
4. How tightly are time and cost linked?<\/h3>\n
A simple example<\/h2>\n
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Why this matters for governance<\/h2>\n
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More honest approvals<\/h3>\n
Better contingency setting<\/h3>\n
Clearer escalation triggers<\/h3>\n
Better portfolio comparison<\/h3>\n
Stronger communication with stakeholders<\/h3>\n
The role of correlation: the part many teams understate<\/h2>\n
Common sources of correlation<\/h3>\n
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Why weak correlation assumptions are risky<\/h3>\n
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Practical guidance<\/h3>\n
What data quality is good enough?<\/h2>\n
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Common modeling mistakes<\/h2>\n
Mistake 1: Treating the schedule as a static date list<\/h3>\n
Mistake 2: Using cost ranges that ignore schedule effects<\/h3>\n
Mistake 3: Double counting risk<\/h3>\n
Mistake 4: Ignoring risk ownership and response plans<\/h3>\n
Mistake 5: Confusing precision with accuracy<\/h3>\n
How to explain the result to non-technical stakeholders<\/h2>\n
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Useful outputs beyond a single confidence number<\/h2>\n
Confidence for any target pair<\/h3>\n
Target pairs for a required confidence level<\/h3>\n
Conditional views<\/h3>\n
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Driver analysis<\/h3>\n
From model to management action<\/h2>\n
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A lightweight implementation approach<\/h2>\n
Step 1: Select the right level of detail<\/h3>\n
Step 2: Identify cost elements that move with time<\/h3>\n
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Step 3: Define uncertainty ranges<\/h3>\n
Step 4: Capture<\/h3>\n
Step 4: Capture the major dependencies<\/h3>\n
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Step 5: Run enough simulations to stabilize the picture<\/h3>\n
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Step 6: Test the approved target and a few alternatives<\/h3>\n
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Step 7: Use the outputs in real decisions<\/h3>\n
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How often the model should be updated<\/h2>\n
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Who should be involved<\/h2>\n
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What a good leadership discussion sounds like<\/h2>\n
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Practical interpretation tips<\/h2>\n
Do not obsess over a single percentage point<\/h3>\n
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Compare scenarios, not just baselines<\/h3>\n
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Focus on the out-of-box drivers<\/h3>\n
Match confidence to the decision type<\/h3>\n
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Limitations to acknowledge<\/h2>\n
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A useful maturity path for organizations<\/h2>\n
Level 1: Separate schedule and cost risk views<\/h3>\n
Level 2: Basic integrated target-box testing<\/h3>\n
Level 3: Explicit dependency modeling<\/h3>\n
Level 4: Scenario-based decision support<\/h3>\n
Level 5: Embedded governance use<\/h3>\n
A concise way to summarize the concept<\/h2>\n
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Final thought<\/h2>\n
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Python Implementation Example<\/h1>\n\n\n\n
import numpy as np\nimport pandas as pd\nimport plotly.express as px\nfrom datetime import datetime, timedelta\n\nnp.random.seed(42)\n\n# ----------------------------\n# Sample integrated project data\n# ----------------------------\nwork_packages = pd.DataFrame({\n \"work_package\": [\"Requirements\", \"Design\", \"Build\", \"Testing\", \"Training\", \"Go-Live Prep\"],\n \"optimistic_days\": [5, 8, 15, 10, 4, 3],\n \"most_likely_days\": [7, 10, 20, 14, 6, 5],\n \"pessimistic_days\": [10, 15, 30, 22, 9, 8],\n \"fixed_cost\": [12000, 20000, 45000, 18000, 8000, 6000],\n \"daily_cost\": [1800, 2500, 4200, 3000, 1500, 1200]\n})\n\nproject_start = datetime(2026, 5, 1)\n\n# ----------------------------\n# Simulation settings\n# ----------------------------\nn_simulations = 10000\ntarget_finish_date = datetime(2026, 6, 30)\ntarget_budget = 260000\n\n# Shared risk drivers to create realistic correlation\n# These affect both duration and cost together\ndriver_names = [\"productivity\", \"vendor_delay\", \"testing_rework\"]\ndriver_std = np.array([0.12, 0.10, 0.15])\ndriver_corr = np.array([\n [1.00, 0.35, 0.25],\n [0.35, 1.00, 0.30],\n [0.25, 0.30, 1.00]\n])\n\n# Cholesky decomposition for correlated random variables\nL = np.linalg.cholesky(driver_corr)\n\n# Exposure of each work package to each shared risk driver\n# Columns: productivity, vendor_delay, testing_rework\ndriver_exposure = {\n \"Requirements\": np.array([0.30, 0.05, 0.00]),\n \"Design\": np.array([0.40, 0.10, 0.05]),\n \"Build\": np.array([0.70, 0.60, 0.20]),\n \"Testing\": np.array([0.35, 0.15, 0.80]),\n \"Training\": np.array([0.20, 0.05, 0.10]),\n \"Go-Live Prep\": np.array([0.20, 0.20, 0.15])\n}\n\ndef sample_correlated_drivers():\n z = np.random.normal(size=len(driver_names))\n correlated = L @ z\n scaled = correlated * driver_std\n return dict(zip(driver_names, scaled))\n\ndef simulate_integrated_model(df, n_runs, start_date):\n results = []\n\n for _ in range(n_runs):\n drivers = sample_correlated_drivers()\n driver_vector = np.array([drivers[\"productivity\"], drivers[\"vendor_delay\"], drivers[\"testing_rework\"]])\n\n total_days = 0\n total_cost = 0\n\n for _, row in df.iterrows():\n base_duration = np.random.triangular(\n row[\"optimistic_days\"],\n row[\"most_likely_days\"],\n row[\"pessimistic_days\"]\n )\n\n exposure = driver_exposure[row[\"work_package\"]]\n combined_driver_effect = np.dot(exposure, driver_vector)\n\n adjusted_duration = max(1, base_duration * (1 + combined_driver_effect))\n\n fixed_cost_shock = np.random.normal(0, row[\"fixed_cost\"] * 0.03)\n adjusted_fixed_cost = max(0, row[\"fixed_cost\"] + fixed_cost_shock)\n\n adjusted_daily_cost = row[\"daily_cost\"] * (1 + 0.6 * combined_driver_effect)\n adjusted_daily_cost = max(0, adjusted_daily_cost)\n\n package_cost = adjusted_fixed_cost + adjusted_duration * adjusted_daily_cost\n\n total_days += adjusted_duration\n total_cost += package_cost\n\n finish_date = start_date + timedelta(days=int(round(total_days)))\n\n results.append({\n \"total_duration_days\": total_days,\n \"finish_date\": finish_date,\n \"total_cost\": total_cost\n })\n\n return pd.DataFrame(results)\n\ndef percentile_finish_date(series, percentile):\n sorted_dates = pd.Series(pd.to_datetime(series)).sort_values().reset_index(drop=True)\n index = int(np.ceil(percentile \/ 100 * len(sorted_dates))) - 1\n return sorted_dates.iloc[max(index, 0)]\n\ndef percentile_cost(series, percentile):\n return np.percentile(series, percentile)\n\ndef joint_confidence(df, target_date, target_cost):\n return ((df[\"finish_date\"] <= pd.to_datetime(target_date)) & (df[\"total_cost\"] <= target_cost)).mean()\n\n# ----------------------------\n# Run simulation\n# ----------------------------\nresults_df = simulate_integrated_model(work_packages, n_simulations, project_start)\n\n# ----------------------------\n# Headline outputs\n# ----------------------------\np50_finish = percentile_finish_date(results_df[\"finish_date\"], 50)\np80_finish = percentile_finish_date(results_df[\"finish_date\"], 80)\np50_cost = percentile_cost(results_df[\"total_cost\"], 50)\np80_cost = percentile_cost(results_df[\"total_cost\"], 80)\n\nfinish_confidence = (results_df[\"finish_date\"] <= pd.to_datetime(target_finish_date)).mean()\ncost_confidence = (results_df[\"total_cost\"] <= target_budget).mean()\ncombined_confidence = joint_confidence(results_df, target_finish_date, target_budget)\n\nprint(\"Integrated Cost and Schedule Risk Results\")\nprint(\"-\" * 50)\nprint(f\"Target finish date: {target_finish_date.strftime('%Y-%m-%d')}\")\nprint(f\"Target budget: ${target_budget:,.0f}\")\nprint(f\"Probability of finishing by target date: {finish_confidence:.1%}\")\nprint(f\"Probability of staying under target budget: {cost_confidence:.1%}\")\nprint(f\"Joint confidence of achieving both: {combined_confidence:.1%}\")\nprint()\nprint(f\"P50 finish date: {p50_finish.strftime('%Y-%m-%d')}\")\nprint(f\"P80 finish date: {p80_finish.strftime('%Y-%m-%d')}\")\nprint(f\"P50 total cost: ${p50_cost:,.0f}\")\nprint(f\"P80 total cost: ${p80_cost:,.0f}\")\n\n# ----------------------------\n# 2D scatter plot of outcomes\n# ----------------------------\nplot_df = results_df.copy()\nplot_df[\"finish_date_str\"] = pd.to_datetime(plot_df[\"finish_date\"]).dt.strftime(\"%Y-%m-%d\")\nplot_df[\"inside_target_box\"] = np.where(\n (plot_df[\"finish_date\"] <= pd.to_datetime(target_finish_date)) &\n (plot_df[\"total_cost\"] <= target_budget),\n \"Inside target\",\n \"Outside target\"\n)\n\nfig_scatter = px.scatter(\n plot_df,\n x=\"total_duration_days\",\n y=\"total_cost\",\n color=\"inside_target_box\",\n opacity=0.55,\n title=\"Integrated Cost and Schedule Risk Outcomes\",\n labels={\n \"total_duration_days\": \"Total Duration (days)\",\n \"total_cost\": \"Total Cost\"\n },\n hover_data=[\"finish_date_str\"]\n)\n\nfig_scatter.add_vline(\n x=(target_finish_date - project_start).days,\n line_dash=\"dash\",\n line_color=\"red\"\n)\n\nfig_scatter.add_hline(\n y=target_budget,\n line_dash=\"dash\",\n line_color=\"red\"\n)\n\nfig_scatter.add_annotation(\n x=(target_finish_date - project_start).days,\n y=target_budget,\n text=\"Target Box Corner\",\n showarrow=True,\n arrowhead=1\n)\n\nfig_scatter.show()\n\n# ----------------------------\n# 3D confidence surface\n# ----------------------------\nduration_grid = np.linspace(\n results_df[\"total_duration_days\"].min(),\n results_df[\"total_duration_days\"].max(),\n 35\n)\n\ncost_grid = np.linspace(\n results_df[\"total_cost\"].min(),\n results_df[\"total_cost\"].max(),\n 35\n)\n\nZ = np.zeros((len(cost_grid), len(duration_grid)))\n\nfor i, cost_limit in enumerate(cost_grid):\n for j, duration_limit in enumerate(duration_grid):\n Z[i, j] = (\n (results_df[\"total_duration_days\"] <= duration_limit) &\n (results_df[\"total_cost\"] <= cost_limit)\n ).mean()\n\nsurface_df = pd.DataFrame(\n Z,\n index=np.round(cost_grid, 0),\n columns=np.round(duration_grid, 1)\n)\n\nfig_surface = px.imshow(\n surface_df,\n origin=\"lower\",\n aspect=\"auto\",\n title=\"Joint Confidence Surface\",\n labels={\n \"x\": \"Duration Threshold (days)\",\n \"y\": \"Cost Threshold\",\n \"color\": \"Confidence\"\n }\n)\n\nfig_surface.add_vline(\n x=np.abs(duration_grid - (target_finish_date - project_start).days).argmin(),\n line_dash=\"dash\",\n line_color=\"red\"\n)\n\nfig_surface.add_hline(\n y=np.abs(cost_grid - target_budget).argmin(),\n line_dash=\"dash\",\n line_color=\"red\"\n)\n\nfig_surface.show()\n\n# ----------------------------\n# Contour chart alternative\n# ----------------------------\nfig_contour = px.density_contour(\n results_df,\n x=\"total_duration_days\",\n y=\"total_cost\",\n nbinsx=30,\n nbinsy=30,\n title=\"Density Contour of Simulated Time and Cost Outcomes\",\n labels={\n \"total_duration_days\": \"Total Duration (days)\",\n \"total_cost\": \"Total Cost\"\n }\n)\n\nfig_contour.add_vline(\n x=(target_finish_date - project_start).days,\n line_dash=\"dash\",\n line_color=\"red\"\n)\n\nfig_contour.add_hline(\n y=target_budget,\n line_dash=\"dash\",\n line_color=\"red\"\n)\n\nfig_contour.show()<\/code><\/pre>\n\n\n\nOutput Example<\/h3>\n\n\n\n
![\"\"](\"https:\/\/i0.wp.com\/hksmnow.com\/project-management\/wp-content\/uploads\/2026\/04\/Integrated-Cost-and-Schedule-Risk-Outcomes.png?resize=1024%2C353&ssl=1\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/i0.wp.com\/hksmnow.com\/project-management\/wp-content\/uploads\/2026\/04\/Joint-Surface.png?resize=1024%2C343&ssl=1\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/i0.wp.com\/hksmnow.com\/project-management\/wp-content\/uploads\/2026\/04\/Desity-Countur.png?resize=1024%2C342&ssl=1\")
<\/figure>\n\n\n\nInterpretation<\/h1>\n\n\n\n
1. Scatter plot: Integrated Cost and Schedule Risk Outcomes<\/h2>\n\n\n\n
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2. Density contour chart<\/h2>\n\n\n\n
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3. Joint confidence surface<\/h2>\n\n\n\n
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px.imshow()<\/code> is using matrix index positions while your red lines are being added in the original value scale, or because the axis labels are not aligned properly with the underlying grid.<\/li>\n<\/ul>\n\n\n\n\n
Overall business interpretation<\/h2>\n\n\n\n
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What to say in project language<\/h2>\n\n\n\n
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Practical takeaway<\/h2>\n\n\n\n
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