29:null 2a:["$","title",null,{"children":"beta-PERT | Christophe Yamahata | Observable"}] 2f:T5ac,// Function to generate a Beta-distributed random value // See: https://observablehq.com/@mbostock/beta-distributed-random-values function betaRand(alpha, beta) { // Returns random values that are normally distributed (mu=0, sigma=1) // See: "Numerical Recipes 3rd Edition: The Art of Scientific Computing," p. 369 function randn() { let u, v, x, y, q; do { u = Math.random(); v = 1.7156 * (Math.random() - 0.5); x = u - 0.449871; y = Math.abs(v) + 0.386595; q = x * x + y * (0.19600 * y - 0.25472 * x); } while (q > 0.27597 && (q > 0.27846 || v * v > -4 * Math.log(u) * u * u)); return v / u; } // Returns a gamma deviate by the method of Marsaglia and Tsang // See: "Numerical Recipes 3rd Edition: The Art of Scientific Computing," p. 370 function randg(shape) { let oalph = shape, a1, a2, u, v, x, mat; if (!shape) shape = 1; if (shape < 1) shape += 1; a1 = shape - 1 / 3; a2 = 1 / Math.sqrt(9 * a1); do { do { x = randn(); v = 1 + a2 * x; } while (v <= 0); v = v * v * v; u = Math.random(); } while ( u > 1 - 0.331 * Math.pow(x, 4) && Math.log(u) > 0.5 * x * x + a1 * (1 - v + Math.log(v)) ); if (shape === oalph) return a1 * v; // alpha > 1 do u = Math.random(); while (u === 0); // alpha < 1 return Math.pow(u, 1 / oalph) * a1 * v; } const u = randg(alpha); return u / (u + randg(beta)); }30:[] 2d:{"id":"6bc8999fc767a289","slug":null,"title":"beta-PERT","thumbnail":"221a8ffbe44586875f43123355e5f757c2156fa3810a3f41ba70789d1c2f3bc4","access_level":"public","create_time":"$D2025-02-03T15:09:17.382Z","update_time":"$D2025-02-07T14:15:46.941Z","view_time":null,"first_public_time":"$D2025-02-07T14:11:49.566Z","fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"dd67891d326bcfb1","login":"christophe-yamahata","name":"Christophe Yamahata","avatar_url":"https://avatars-next.observableusercontent.com/8256ac716aa8395babba412b2835913d186874fa4190df57a4a28f2e60129876","tier":"starter_2024"},"version":1166,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"06f083eba60f5a76","login":"christophe-yamahata","name":"Christophe Yamahata","avatar_url":"https://avatars-next.observableusercontent.com/8256ac716aa8395babba412b2835913d186874fa4190df57a4a28f2e60129876","tier":null}],"members":[{"type":"individual","id":"06f083eba60f5a76","login":"christophe-yamahata","name":"Christophe Yamahata","avatar_url":"https://avatars-next.observableusercontent.com/8256ac716aa8395babba412b2835913d186874fa4190df57a4a28f2e60129876","tier":null,"role":"editor"},{"type":"team","id":"dd67891d326bcfb1","login":"christophe-yamahata","name":"Christophe Yamahata","avatar_url":"https://avatars-next.observableusercontent.com/8256ac716aa8395babba412b2835913d186874fa4190df57a4a28f2e60129876","tier":"starter_2024","role":"editor"}],"comments":[],"custom_thumbnail":true,"trashed":false,"subscribed":null,"forks":0,"collection":{"id":"a4317cf9df48c2f1","slug":"sandbox","title":"Sandbox","owner":{"id":"dd67891d326bcfb1","login":"christophe-yamahata","name":"Christophe Yamahata"}},"type":"notebook","body":{"title":"beta-PERT","cells":[{"id":0,"value":"# beta-PERT","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1048,"value":"This basic notebook is placed in the ‘sandbox’ to explore the beta-BERT function used in Monte Carlo schedule analysis. It serves as an initial step towards the detailed notebooks and more advanced tools I have developed on schedule analysis, notably the PERT-based Monte Carlo simulations.\n

","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":739,"value":"## PERT\n\nThe PERT distribution considers three key estimates for a task: the **Optimistic (O)**, **Most Likely (ML)**, and **Pessimistic (P)** durations.\n
","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":260,"value":"// Optimized input handling\nviewof Optimistic = Inputs.range([0, 99.8], { value: 0, step: 0.1, label: \"Optimistic (O)\" });\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":266,"value":"viewof mostLikely = {\n let mostLikelyValue = 0.25 * (Optimistic + Pessimistic);\n\nlet mostLikely = Inputs.range([\n Optimistic + 0.08 * (Pessimistic - Optimistic),\n Pessimistic - 0.08 * (Pessimistic - Optimistic)\n], { \n value: mostLikelyValue, \n step: 0.1, \n label: \"Most Likely (ML)\"\n});\n\nlet clonedMostLikely = Inputs.range([\n Optimistic + 0.08 * (Pessimistic - Optimistic),\n Pessimistic - 0.08 * (Pessimistic - Optimistic)\n], { \n value: mostLikelyValue, \n step: 0.1, \n label: \"Cloned Most Likely (ML)\"\n});\n\n// Sync both sliders\nmostLikely.addEventListener('input', (event) => {\n clonedMostLikely.value = event.target.value;\n});\n return mostLikely\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":264,"value":"viewof Pessimistic = Inputs.range([Optimistic + 0.2, 100], { value: 100, step: 0.1, label: \"Pessimistic (P)\" });\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1056,"value":"
\nUsing the values above, one can calculate the expected duration (μ), variance (σ²), and standard deviation (σ) to model the uncertainty of an activity duration in project scheduling:","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":694,"value":"md`\nExpected duration: ${tex`\\mu = \\frac{O + 4ML + P}{6} = ${((Optimistic + 4 * mostLikely + Pessimistic) / 6).toFixed(2)}`}\n
\nVariance: ${tex`\\sigma^2 = \\frac{(P - O)^2}{36} = ${((Math.pow(Pessimistic - Optimistic, 2)) / 36).toFixed(2)}`}\n
\nStandard deviation: ${tex`\\sigma = ${Math.pow(Math.pow(Pessimistic - Optimistic, 2) / 36, 0.5).toFixed(2)}`}\n

\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":737,"value":"## beta-PERT Distribution\n\nThe beta-PERT distribution for the given parameters is visualized in the chart (solid line). The histogram, derived from ${length.toLocaleString()} random data points, illustrates the empirical outcome, typically produced by running a Monte Carlo simulation. \n\nBy adjusting the “Most Likely” slider, you can observe how changes to this parameter influence the shape of the distribution.\n

\n","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":21,"value":"plot = Plot.plot({\n y: { grid: true },\n x: { domain: [Optimistic, Pessimistic], label: \"Duration\" },\n y: { domain: [0, Math.max(length * 2.5 / bins.length, Math.ceil((maxCount / 1000) * 1000))] },\n marks: [\n Plot.rectY(dataRND, Plot.binX({ y: \"count\" }, { x: \"value\", fill: \"steelblue\" })),\n Plot.line(betaData, { x: \"x\", y: d => d.y, curve: \"catmull-rom\", stroke: \"#000\", strokeWidth: 5, opacity: 0.3 }),\n Plot.ruleY([0])\n ]\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":32,"value":"viewof updateButton = Inputs.button(html`${isRunning ? 'STOP' : 'Live update'}`, {\n label: html`Random data`,\n reduce: () => {\n if (mutable isRunning) {\n clearInterval(mutable intervalId);\n mutable isRunning = false;\n } else {\n mutable isRunning = true;\n mutable intervalId = setInterval(() => {\n if (!mutable isRunning) clearInterval(mutable intervalId);\n mutable dataRND = Array.from({ length }, () => ({ value: betaPERT(Optimistic, mostLikely, Pessimistic) }));\n }, 50);\n }\n }\n});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1116,"value":"clonedMostLikely = {\n const slider = Inputs.range([\n Optimistic + 0.08 * (Pessimistic - Optimistic),\n Pessimistic - 0.08 * (Pessimistic - Optimistic)\n ], { \n disabled: true,\n value: mostLikely, \n step: 0.1, \n label: \"Most Likely (ML)\" \n });\n \n const percentage = `${((mostLikely-Optimistic)*100/(Pessimistic - Optimistic)).toFixed(0)}%`;\n \n return html`

${slider}${percentage}

`;\n}\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1152,"value":"
\n\n---","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined"},{"id":1144,"value":"viewof display_inspect = Inputs.toggle({label: \"⇣ Check to inspect variables and functions\", values: [\"true\", \"none\"], value: \"false\" })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":512,"value":"\n
\n${display_inspect == \"none\"? html`

⇣ Appendix is hidden ...

` : md`## Appendix`}","pinned":false,"hidden":false,"mode":"html","author":"$undefined","summary":"$undefined"},{"id":235,"value":"length = 10000;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":175,"value":"mutable isRunning = {\n mostLikely;\n return false;\n} ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":194,"value":"mutable intervalId = 0","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":4,"value":"mutable dataRND = Array.from({ length }, () => ({ value: betaPERT(Optimistic, mostLikely, Pessimistic) }));","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":599,"value":"// Histogram calculation\nbins = d3.bin()\n .value(d => d.value)\n .domain([Optimistic, Pessimistic])\n .thresholds(d3.thresholdScott)(dataRND);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":615,"value":"maxCount = Math.max(...bins.map(bin => bin.length));","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":755,"value":"betaData = {\n let betaData = normalizeBetaPERT(\n Optimistic, \n mostLikely, \n Pessimistic, \n bins.map(bin => ({ x: bin.x0, count: bin.length }))\n );\n \n // Ensure the last point is included\n betaData.push({ x: Pessimistic, y: 0 });\n return betaData;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":9,"value":"$2f","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":746,"value":"// Gamma function (used in the Beta function)\nfunction gamma(x) {\n if (x === 0) return Infinity;\n let z = 1;\n for (let i = 2; i < x; i++) z *= i;\n return z;\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1005,"value":"alpha = computeBetaPERTParams(Optimistic,mostLikely,Pessimistic).alpha;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":1023,"value":"beta = computeBetaPERTParams(Optimistic,mostLikely,Pessimistic).beta;","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":946,"value":"// beta-PERT distribution computation\nfunction computeBetaPERTParams(O, ML, P) {\n const mu = (O + 4 * ML + P) / 6;\n const sigma = (P - O) / 6;\n const alpha = ((mu - O) / (P - O)) * (((mu - O) * (P - mu)) / (sigma * sigma) - 1);\n const beta = (alpha * (P - mu)) / (mu - O);\n return { alpha, beta };\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":948,"value":"function betaPERT(O, ML, P) {\n return O + betaRand(alpha, beta) * (P - O);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":955,"value":"// beta-PERT PDF\nfunction betaPERTPDF(O, ML, P, x, alpha, beta) {\n const betaFunc = (gamma(alpha) * gamma(beta)) / gamma(alpha + beta);\n return (Math.pow(x - O, alpha - 1) * Math.pow(P - x, beta - 1)) /\n (Math.pow(P - O, alpha + beta - 1) * betaFunc);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"},{"id":801,"value":"// beta-PERT PDF normalization\nfunction normalizeBetaPERT(O, ML, P, data) {\n const maxPDF = Math.max(...data.map(d => betaPERTPDF(O, ML, P, d.x, alpha, beta)));\n const scalingFactor = maxCount / maxPDF;\n return data.map(d => ({ x: d.x, y: betaPERTPDF(O, ML, P, d.x, alpha, beta) * scalingFactor }));\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined"}],"files":"$W30","stdlib":"1","titling":"auto"}} 2e:"$2d:body"