2c:null 2d:["$","title",null,{"children":"Inventory management | Aditya Mahajan | Observable"}] 32:T83e,binomial = { // From https://stackoverflow.com/a/37715980/193149 const logf = [0, 0, 0.6931471805599453, 1.791759469228055, 3.1780538303479458, 4.787491742782046, 6.579251212010101, 8.525161361065415, 10.60460290274525, 12.801827480081469, 15.104412573075516, 17.502307845873887, 19.987214495661885, 22.552163853123425, 25.19122118273868, 27.89927138384089, 30.671860106080672, 33.50507345013689, 36.39544520803305, 39.339884187199495, 42.335616460753485, 45.38013889847691, 48.47118135183523, 51.60667556776438, 54.78472939811232, 58.00360522298052, 61.261701761002, 64.55753862700634, 67.88974313718154, 71.25703896716801, 74.65823634883016, 78.0922235533153, 81.55795945611504, 85.05446701758152, 88.58082754219768, 92.1361756036871, 95.7196945421432, 99.33061245478743, 102.96819861451381, 106.63176026064346, 110.32063971475739, 114.0342117814617, 117.77188139974507, 121.53308151543864, 125.3172711493569, 129.12393363912722, 132.95257503561632, 136.80272263732635, 140.67392364823425, 144.5657439463449, 148.47776695177302, 152.40959258449735, 156.3608363030788, 160.3311282166309, 164.32011226319517, 168.32744544842765, 172.3527971391628, 176.39584840699735, 180.45629141754378, 184.53382886144948, 188.6281734236716, 192.7390472878449, 196.86618167289, 201.00931639928152, 205.1681994826412, 209.34258675253685, 213.53224149456327, 217.73693411395422, 221.95644181913033, 226.1905483237276, 230.43904356577696, 234.70172344281826, 238.97838956183432, 243.2688490029827, 247.57291409618688, 251.8904022097232, 256.22113555000954, 260.5649409718632, 264.9216497985528, 269.2910976510198, 273.6731242856937, 278.0675734403661, 282.4742926876304, 286.893133295427, 291.3239500942703, 295.76660135076065, 300.22094864701415, 304.6868567656687, 309.1641935801469, 313.65282994987905, 318.1526396202093, 322.66349912672615, 327.1852877037752, 331.7178871969285, 336.26118197919845, 340.815058870799, 345.37940706226686, 349.95411804077025, 354.5390855194408, 359.1342053695754, 363.73937555556347] return function(n, k, p) { return Math.exp(logf[n] - logf[n-k] - logf[k]) * p**k * (1-p)**(n-k) } }33:T46d,DP = { var DP = new Array() var idx = 0 const Smax = (W.length - 1)*T var V = new Array(2*Smax + 1) // -Smax to Smax var H = new Array(3*Smax + 1) // -Smax to 2Smax // Initialize the terminal value function for( var s = -Smax; s <= Smax; s++) { V[s + Smax] = 0 } // Dynamic programming for( var t = T; t >= 0; t--) { // H(z) = EXP[ h(z-W) + V(z-W) ] for( var z = -Smax; z <= 2*Smax; z++) { H[z + Smax] = 0 for( var i = 0; i < W.length; i++ ) { var w = W[i].demand H[z + Smax] += (h(z - w) + V[Math.min(Math.max(z-w, -Smax), Smax) + Smax])*W[i].probability } } // V(s) = min_{a in S_{>= 0}} { p*a + H(s+a) } for(var s = -Smax; s <= Smax; s++) { var opt = 0 var val = H[s + Smax] for(var a = 1; a <= Smax; a++) { var newVal = p*a + H[Math.min(s+a, 2*Smax) + Smax ] if(newVal <= val) { opt = a val = newVal } } V[s + Smax] = val DP[idx++] = { time: t, state: s, value: val, action: opt } } } return DP }34:T4f7,infDP = { var infDP = new Array() var idx = 0 const Titerations = 50// Number of iterations const Smax = (W.length - 1)*Titerations var V = new Array(2*Smax + 1) // -Smax to Smax var H = new Array(3*Smax + 1) // -Smax to 2Smax // Initialize the terminal value function for( var s = -Smax; s <= Smax; s++) { V[s + Smax] = 0 } // Dynamic programming for( var t = Titerations; t >= 0; t--) { // H(z) = EXP[ h(z-W) + V(z-W) ] for( var z = -Smax; z <= 2*Smax; z++) { H[z + Smax] = 0 for( var i = 0; i < W.length; i++ ) { var w = W[i].demand H[z + Smax] += (h(z - w) + V[Math.min(Math.max(z-w, -Smax), Smax) + Smax])*W[i].probability } } // V(s) = min_{a in S_{>= 0}} { p*a + discount*H(s+a) } for(var s = -Smax; s <= Smax; s++) { var opt = 0 var val = discount*H[s + Smax] for(var a = 1; a <= Smax; a++) { var newVal = p*a + discount*H[Math.min(s+a, 2*Smax) + Smax ] if(newVal <= val) { opt = a val = newVal } } V[s + Smax] = val if (t==0){ infDP[idx++] = { time: t, state: s, value: val, action: opt } } } } return infDP }35:[] 30:{"id":"b2eb5b0d068edd4a","slug":"inventory-management","title":"Inventory management","thumbnail":"8e70e5d1b494ee09fa165e8923f85a58788380f51156d76990efa6310d6bd5be","access_level":"public","create_time":"$D2022-01-13T07:25:28.838Z","update_time":"$D2022-10-04T00:50:41.626Z","view_time":null,"first_public_time":"$D2023-03-22T01:22:19.017Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"ff25aa15160a9464","login":"adityam","name":"Aditya Mahajan","avatar_url":"https://avatars-next.observableusercontent.com/5fb86fe7ae5ef2cc6f848cd5b35a5ce133609711914aff33ba713d95263439ca","tier":"starter_2024"},"version":238,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"bb2fa893d524052c","login":"adityam","name":"Aditya Mahajan","avatar_url":"https://avatars-next.observableusercontent.com/5fb86fe7ae5ef2cc6f848cd5b35a5ce133609711914aff33ba713d95263439ca","tier":null}],"members":[{"type":"individual","id":"bb2fa893d524052c","login":"adityam","name":"Aditya Mahajan","avatar_url":"https://avatars-next.observableusercontent.com/5fb86fe7ae5ef2cc6f848cd5b35a5ce133609711914aff33ba713d95263439ca","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":2,"collection":null,"type":"notebook","body":{"titling":"auto","title":"Inventory management","cells":[{"id":0,"value":"# Inventory management","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":7,"value":"viewof p = Inputs.range([1, 15], {label: \"Purchase cost\", step: 1, value: 5 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":11,"value":"viewof ch = Inputs.range([1, 10], {label: \"Holding cost\", step: 1, value: 4 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":16,"value":"viewof cs = Inputs.range([1, 10], {label: \"Shortage cost\", step: 1, value: 2 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":70,"value":"h = function(s) {\n if(s >= 0) \n return ch*s\n else\n return -cs*s\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":38,"value":"## Demand\n\nWe assume that the demand is a Binomial distribution Bin(10,0.4)","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":42,"value":"demand_plot = Plot.plot({\n grid: true,\n marks: [\n // Axes\n Plot.ruleX([0]),\n Plot.ruleY([0]),\n // Data\n Plot.line(W, {x:\"demand\", y:\"probability\",curve:\"step-after\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":29,"value":"W = {\n const s = 0.4\n const n = 10\n \n var points = new Array(n+1)\n var cumulative = 0\n var probability = 0\n\n for(var k = 0; k <= n; k++) {\n probability = binomial(n,k,s)\n cumulative += probability\n points[k] = { demand:k, probability: probability, cumulative: cumulative }\n }\n\n return points\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":21,"value":"$32","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"## Computation of Value functions","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":62,"value":"T = 15","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":138,"value":"viewof time = Inputs.range([1, T], {label: \"T\", step: 1, value: T })","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":96,"value":"value_plot = Plot.plot({\n grid: true,\n marks: [\n // Axes\n Plot.ruleX([0]),\n Plot.ruleY([0]),\n // Data\n Plot.line(DP.filter(d => d.time == time && Math.abs(d.state) <= 25), {x:\"state\", y:\"value\", curve:\"step-after\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":124,"value":"action_plot = Plot.plot({\n grid: true,\n marks: [\n // Axes\n Plot.ruleX([0]),\n Plot.ruleY([0]),\n // Data\n Plot.line(DP.filter(d => d.time == time && Math.abs(d.state) <= 20), {x:\"state\", y:\"action\", curve:\"step-after\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":66,"value":"$33","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":101,"value":"DP.filter(d => d.time==time)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":198,"value":"## Infinite horizon setting","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":168,"value":"viewof discount = Inputs.range([0.75, 0.9], {label: \"Discount factor\", step: 0.05, value: 0.9 })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":175,"value":"inf_value_plot = Plot.plot({\n grid: true,\n marks: [\n // Axes\n Plot.ruleX([0]),\n Plot.ruleY([0]),\n // Data\n Plot.line(infDP.filter(d => Math.abs(d.state) <= 25), {x:\"state\", y:\"value\", curve:\"step-after\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":185,"value":"inf_action_plot = Plot.plot({\n grid: true,\n marks: [\n // Axes\n Plot.ruleX([0]),\n Plot.ruleY([0]),\n // Data\n Plot.line(infDP.filter(d => Math.abs(d.state) <= 20), {x:\"state\", y:\"action\", curve:\"step-after\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":191,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":224,"value":"## Optimal threshold","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":228,"value":"F_opt = (cs - p*(1-discount)/discount)/(ch + cs)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":226,"value":"demand_CDF_plot = Plot.plot({\n grid: true,\n marks: [\n // Axes\n Plot.ruleX([0]),\n Plot.ruleY([0]),\n // Data\n Plot.line(W, {x:\"demand\", y:\"cumulative\",curve:\"step-after\"}),\n Plot.line([ [0,F_opt], [W[W.length-1].demand, F_opt] ], {stroke: \"red\"})\n ]\n})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W35","stdlib":"1"}} 31:"$30:body"