{ "cells": [ { "cell_type": "markdown", "id": "5050a488-083e-4637-ad53-ded95e92a861", "metadata": {}, "source": [ "Math 420/620 Project" ] }, { "cell_type": "markdown", "id": "c99779a6-7512-49c6-a0bb-e5f780baf170", "metadata": {}, "source": [ "A mechanic working for a heavy equipment repair facility is\n", "responsible for the repair and maintenance of forklift trucks.\n", "When forklifts break down they are taken to the repair facility\n", "and serviced in the order of their arrival.\n", "Assume rate at which forklifts break down \n", "is $\\lambda$ per month\n", "and the rate they can be repaired \n", "is $\\mu$ per month.\n", "Since there is space for 27 forklifts at the facility,\n", "the number of forklifts in repair at time $t$ can be\n", "represented by\n", "$$X_t\\in \\{ 0, 1, 2, \\ldots, 27\\}.$$\n", "The only possible transitions are from $X_t=i$ to $X_t=i+1$ or $i-1$.\n", "Thus, the transition rate diagram looks like\n" ] }, { "attachments": { 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FRcVNmzZBuYuRpPR1cXF5+PDhsD8/n88/cOCAmpqaiorK9u3bodzFqLy8/LXX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} }, "cell_type": "markdown", "id": "cdd8362b-9ab9-4720-8d5c-594133808ba0", "metadata": {}, "source": [ "![forkrate.png](attachment:56fc9623-c209-4bb3-b4ee-bf01e6135d8c.png)" ] }, { "cell_type": "markdown", "id": "be8c2cff-9355-4420-b109-01c012b1fc6c", "metadata": {}, "source": [ "and the probability distribution\n", "$P(t)=\\big(P_0(t),P_1(t),P_2(t),\\ldots,P_{27}(t)\\big)\n", "$ \n", "of the corresponding\n", "Markov process satisfies\n", "$$\n", " {dP\\over dt} = P A\n", "\\qquad\\hbox{where}\\qquad\n", " A=\\left[\\matrix{ -\\lambda&\\lambda&&&0\\cr\n", " \\mu & -\\mu-\\lambda&\\ddots&&\\cr\n", " & \\mu & \\ddots&\\lambda&\\cr\n", " && \\ddots&-\\mu-\\lambda&\\lambda\\cr\n", " 0&&& \\mu & -\\mu\\cr\n", " }\\right].\n", "$$\n", "Note that $A\\in{\\bf R}^{28\\times 28}$ and $P\\in{\\bf R}^{1\\times 28}$ is a row vector.\n" ] }, { "cell_type": "markdown", "id": "b9e23943-e1ad-4505-82e5-db284168bd05", "metadata": {}, "source": [ "Note that answers will be different\n", "with the version because each project version has\n", "different values of $\\lambda$ and $\\mu$. As a\n", "result we create a list of those indexed by version number here." ] }, { "cell_type": "code", "execution_count": 1, "id": "7cd98b0d-562f-4318-9e24-5587cea34b90", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "┌─────────┬────────┬─────┐\n", "│\u001b[1m version \u001b[0m│\u001b[1m lambda \u001b[0m│\u001b[1m mu \u001b[0m│\n", "├─────────┼────────┼─────┤\n", "│ 1 │ 4.9 │ 7.4 │\n", "│ 2 │ 4.8 │ 7.7 │\n", "│ 4 │ 3.7 │ 5.6 │\n", "│ 5 │ 4.8 │ 6.8 │\n", "│ 6 │ 4.9 │ 7.3 │\n", "│ 7 │ 5.2 │ 8.3 │\n", "│ 8 │ 3.6 │ 7.0 │\n", "│ 9 │ 5.5 │ 7.4 │\n", "│ 10 │ 4.2 │ 6.1 │\n", "└─────────┴────────┴─────┘\n" ] } ], "source": [ "Ls=[4.9,4.8,3.7,4.8,4.9,5.2,3.6,5.5,4.2]\n", "Ms=[7.4,7.7,5.6,6.8,7.3,8.3,7.0,7.4,6.1]\n", "Ps=[1,2,4,5,6,7,8,9,10]\n", "Is=1:length(Ps)\n", "using PrettyTables\n", "T1=[Ps Ls Ms]\n", "pretty_table(T1,\n", " formatters=[fmt__printf(\"%3d\",[1]),fmt__printf(\"%.1f\",[2,3])],\n", " column_labels=[\"version\",\"lambda\",\"mu\"])" ] }, { "cell_type": "markdown", "id": "70aee3e5-1693-401b-8269-e006d09d49dd", "metadata": {}, "source": [ "(i) Solve $PA=0$ where $P_i\\ge 0$ and\n", "$\\sum_{i=0}^{27} {P_i}=1$\n", "to obtain the steady-state distribution $P=(P_0,P_1,P_2,\\ldots,P_{27})$ \n", "such that $P_i={\\bf P}(X_t=i)$ in the limit." ] }, { "cell_type": "code", "execution_count": 2, "id": "0bf9b178-57b4-4dec-96ae-35bb539ce742", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "mkA (generic function with 1 method)" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "using LinearAlgebra\n", "# Given lambda and mu make the tridiagonal matrix\n", "function mkA(lambda,mu)\n", " a1=mu*ones(27)\n", " a3=lambda*ones(27)\n", " a2=[-lambda,-(mu+lambda)ones(26)...,-mu]\n", " return Tridiagonal(a1,a2,a3)\n", "end" ] }, { "cell_type": "code", "execution_count": 3, "id": "553abfac-7c15-40b0-a4f9-7e6344d8d963", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "28×28 Tridiagonal{Float64, Vector{Float64}}:\n", " -4.9 4.9 ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅ ⋅ \n", " 7.4 -12.3 4.9 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \n", " ⋅ 7.4 -12.3 4.9 ⋅ ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ 7.4 -12.3 4.9 ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ 7.4 -12.3 ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ 7.4 … ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \n", " ⋮ ⋱ ⋮ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ 4.9 ⋅ ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ -12.3 4.9 ⋅ ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ … 7.4 -12.3 4.9 ⋅ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 7.4 -12.3 4.9\n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 7.4 -7.4" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Example matrix for version 1. Set COLUMNS so no wrapping later. \n", "ENV[\"COLUMNS\"]=72\n", "A=mkA(Ls[1],Ms[1])" ] }, { "cell_type": "code", "execution_count": 4, "id": "e02d86e5-758d-4e0e-8602-f5667b618efe", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "mkP0 (generic function with 1 method)" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Find the steady state\n", "function mkP0(A)\n", " K28=eigvecs(A')[:,28]\n", " return K28/sum(K28)\n", "end" ] }, { "cell_type": "code", "execution_count": 5, "id": "34564e64-bec9-415a-8020-306a47a2a42d", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "┌──────────┬───────────────────┬──────────┬───────────────────┐\n", "│\u001b[1m n \u001b[0m│\u001b[1m P0(n) \u001b[0m│\u001b[1m n \u001b[0m│\u001b[1m P0(n) \u001b[0m│\n", "├──────────┼───────────────────┼──────────┼───────────────────┤\n", "│ 0 │ 0.337841116635315 │ 14 │ 0.001052479263803 │\n", "│ 1 │ 0.223705604258519 │ 15 │ 0.000696911944950 │\n", "│ 2 │ 0.148129386603614 │ 16 │ 0.000461468720305 │\n", "│ 3 │ 0.098085674913204 │ 17 │ 0.000305567125607 │\n", "│ 4 │ 0.064948622577662 │ 18 │ 0.000202334988578 │\n", "│ 5 │ 0.043006520355479 │ 19 │ 0.000133978573518 │\n", "│ 6 │ 0.028477290505655 │ 20 │ 0.000088715541924 │\n", "│ 7 │ 0.018856584253744 │ 21 │ 0.000058744075058 │\n", "│ 8 │ 0.012486116600453 │ 22 │ 0.000038898103754 │\n", "│ 9 │ 0.008267833965164 │ 23 │ 0.000025756852486 │\n", "│ 10 │ 0.005474646814771 │ 24 │ 0.000017055213133 │\n", "│ 11 │ 0.003625103971943 │ 25 │ 0.000011293316804 │\n", "│ 12 │ 0.002400406684124 │ 26 │ 0.000007478007073 │\n", "│ 13 │ 0.001589458480028 │ 27 │ 0.000004951653332 │\n", "└──────────┴───────────────────┴──────────┴───────────────────┘\n" ] } ], "source": [ "P0=mkP0(A)\n", "T2=[0:13 P0[1:14] 14:27 P0[15:28]]\n", "pretty_table(T2,\n", " formatters=[fmt__printf(\"%8d\",[1,3]),fmt__printf(\"%.15f\",[2,4])],\n", " column_labels=[\"n\",\"P0(n)\",\"n\",\"P0(n)\"])" ] }, { "cell_type": "markdown", "id": "9d7c79e5-3248-444d-97d7-74fe9edcc067", "metadata": {}, "source": [ "Remark that the above table and following histogram\n", "are for version 1." ] }, { "cell_type": "code", "execution_count": 6, "id": "f7b7d5b1-a5ee-46db-bb2f-5afb7a7a2d7f", "metadata": {}, "outputs": [ { "data": { "image/png": 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"image/svg+xml": [ "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", " \n", " \n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ], "text/html": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "using Plots\n", "bar(0:27,P0,legend=:false,\n", " xlabel=\"forklifts waiting for repair\",ylabel=\"probability\")" ] }, { "cell_type": "markdown", "id": "b54c4516-718c-4138-aac5-238d4fd58c2d", "metadata": {}, "source": [ "(ii) Compute percentage ${\\bf P}(X_t=0)$ of time the mechanic does\n", "not have work." ] }, { "cell_type": "markdown", "id": "36427ac8-d126-4c20-8b85-ba2fdb97fcea", "metadata": {}, "source": [ "Assuming no forklifts are broken at time $t=0$ the\n", "For convenience convert everything to column vectors\n", "by writing $w=P^T$ and $B=A^T$. The differential equation\n", "involving the evolution of $w$ is $dw/dt=Bw$.\n", "\n", "Suppose $K_i$ are the eigenvectors of $B$ with corresponding\n", "eigenvalues $\\alpha_i$. Then the\n", "general solution is of the form\n", "$$\n", " w(t)=c_1K_1e^{\\alpha_1t}+c_2K_2e^{\\alpha_2t}+\\cdots+c_{28}K_{28}e^{\\alpha_{28}t}.\n", "$$\n", "Assume no forklifts are in for repair at time $t=0$. Therefore\n", "$$\n", " w(0)=(1,0,\\ldots,0)=e_1.\n", "$$" ] }, { "cell_type": "code", "execution_count": 7, "id": "388f596f-cf70-43ba-82cd-e059ccb1411d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "w (generic function with 1 method)" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "B=A'\n", "K=eigvecs(B)\n", "alpha=eigvals(B)\n", "e1=I(28)[:,1]\n", "c=K\\e1\n", "w(t)=sum(c[i]*K[:,i]*exp(alpha[i]*t) for i=1:28)" ] }, { "cell_type": "code", "execution_count": 8, "id": "a2b05828-92cd-42da-9a02-a8f6fb4ab688", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "w0 (generic function with 1 method)" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# w0(t) is the probability the mechanic doesn't have work at time t\n", "w0(t)=w(t)[1]" ] }, { "cell_type": "code", "execution_count": 9, "id": "32051915-986d-40c2-8900-fa622919b820", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.9999999999999998" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Up to rounding no work to start with\n", "w0(0)" ] }, { "cell_type": "code", "execution_count": 10, "id": "475302f1-740e-470f-a8f5-330013a1c1f0", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "0.33784111663531485" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# The equilibrium probablity there is no work\n", "P0[1]" ] }, { "cell_type": "markdown", "id": "3d8498e4-dcec-46af-bf8c-96b5bcfbbc67", "metadata": {}, "source": [ "We know $w$ converges to the equilibrium state. Thus,\n", "$$ w(t)\\to P_0 \\approx 0.33784\\qquad\\hbox{as}\\qquad t\\to\\infty.$$" ] }, { "cell_type": "code", "execution_count": 11, "id": "d7873dd0-f71e-4acd-9916-afa386a37c8d", "metadata": {}, "outputs": [ { "data": { "image/png": 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", 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"outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "┌─────────┬────────┬──────┬───────────────────┐\n", "│\u001b[1m version \u001b[0m│\u001b[1m lambda \u001b[0m│\u001b[1m mu \u001b[0m│\u001b[1m P(Xt=0) \u001b[0m│\n", "├─────────┼────────┼──────┼───────────────────┤\n", "│ 1 │ 4.90 │ 7.40 │ 0.337841116635315 │\n", "│ 2 │ 4.80 │ 7.70 │ 0.376624051030879 │\n", "│ 4 │ 3.70 │ 5.60 │ 0.339288811370224 │\n", "│ 5 │ 4.80 │ 6.80 │ 0.294134749236972 │\n", "│ 6 │ 4.90 │ 7.30 │ 0.328771793586915 │\n", "│ 7 │ 5.20 │ 8.30 │ 0.373494745373329 │\n", "│ 8 │ 3.60 │ 7.00 │ 0.485714289696327 │\n", "│ 9 │ 5.50 │ 7.40 │ 0.256820041068969 │\n", "│ 10 │ 4.20 │ 6.10 │ 0.311484429307606 │\n", "└─────────┴────────┴──────┴───────────────────┘\n" ] } ], "source": [ "P0s=[mkP0(mkA(Ls[i],Ms[i]))[1] for i=Is]\n", "T3=[Ps Ls Ms P0s]\n", "pretty_table(T3,\n", " formatters=[fmt__printf(\"%3d\",[1]),\n", " fmt__printf(\"%.2f\",[2,3]),fmt__printf(\"%.15f\",[4])],\n", " column_labels=[\"version\",\"lambda\",\"mu\",\"P(Xt=0)\"])" ] }, { "cell_type": "markdown", "id": "8c25abbe-2b1d-49bc-8665-aeb48c95bc2c", "metadata": {}, "source": [ "(iii) Compute the expected queue length ${\\bf E}[X_t]$ of forklifts\n", "in the repair facility." ] }, { "cell_type": "markdown", "id": "598c012e-b19d-4c9f-9e7a-29aaa489966b", "metadata": {}, "source": [ "By definition\n", "$$\n", " {\\bf E}[X_t]=\\sum_{n=0}^{27} n {\\bf P}(X_t=n).\n", "$$\n", "For large $t$ this reflects the equilibrium state. Thus,\n", "$$\n", " {\\bf E}[X_t]\\approx \\lim_{t\\to\\infty} {\\bf E}[X_t] = \\sum_{n=0}^{27} n P_n.\n", "$$" ] }, { "cell_type": "code", "execution_count": 13, "id": "bc5689b9-3ed1-4b61-8305-bfeed4618de2", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "┌─────────┬────────┬──────┬────────────┬────────────┐\n", "│\u001b[1m version \u001b[0m│\u001b[1m lambda \u001b[0m│\u001b[1m mu \u001b[0m│\u001b[1m P(Xt=0) \u001b[0m│\u001b[1m E[Xt] \u001b[0m│\n", "├─────────┼────────┼──────┼────────────┼────────────┤\n", "│ 1 │ 4.90 │ 7.40 │ 0.33784112 │ 1.95972825 │\n", "│ 2 │ 4.80 │ 7.70 │ 0.37662405 │ 1.65512228 │\n", "│ 4 │ 3.70 │ 5.60 │ 0.33928881 │ 1.94711283 │\n", "│ 5 │ 4.80 │ 6.80 │ 0.29413475 │ 2.39837187 │\n", "│ 6 │ 4.90 │ 7.30 │ 0.32877179 │ 2.04126891 │\n", "│ 7 │ 5.20 │ 8.30 │ 0.37349475 │ 1.67736167 │\n", "│ 8 │ 3.60 │ 7.00 │ 0.48571429 │ 1.05882330 │\n", "│ 9 │ 5.50 │ 7.40 │ 0.25682004 │ 2.88783552 │\n", "│ 10 │ 4.20 │ 6.10 │ 0.31148443 │ 2.20971551 │\n", "└─────────┴────────┴──────┴────────────┴────────────┘\n" ] } ], "source": [ "function getE(i)\n", " P=mkP0(mkA(Ls[i],Ms[i]))\n", " # remark the P[n] is stored off by one\n", " return sum(n*P[n+1] for n=0:27)\n", "end\n", "Es=getE.(Is)\n", "T4=[Ps Ls Ms P0s Es]\n", "pretty_table(T4,\n", " formatters=[fmt__printf(\"%3d\",[1]),\n", " fmt__printf(\"%.2f\",[2,3]),fmt__printf(\"%.8f\",[4,5])],\n", " column_labels=[\"version\",\"lambda\",\"mu\",\"P(Xt=0)\",\"E[Xt]\"])" ] }, { "cell_type": "markdown", "id": "b2f106c3-4660-4611-ae96-304914b50050", "metadata": {}, "source": [ "(iii) What percentage of the time\n", "are 5 or more forklifts in the facility?" ] }, { "cell_type": "markdown", "id": "9f53913b-fb46-4611-a0bf-e80a17528eb2", "metadata": {}, "source": [ "To compute $P(X_t\\ge 5)$ again assume $t$ is so\n", "the probabilities reflect the equilibrium state. Thus\n", "$$\n", " P(X_t\\ge 5)\\approx \\sum_{n=5}^{27} P_n.\n", "$$" ] }, { "cell_type": "code", "execution_count": 14, "id": "39f25f18-bfa5-401d-8b66-725dd3962b69", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "┌─────────┬────────┬──────┬───────────────────┐\n", "│\u001b[1m version \u001b[0m│\u001b[1m lambda \u001b[0m│\u001b[1m mu \u001b[0m│\u001b[1m P(Xt>=5) \u001b[0m│\n", "├─────────┼────────┼──────┼───────────────────┤\n", "│ 1 │ 4.90 │ 7.40 │ 0.127289595011686 │\n", "│ 2 │ 4.80 │ 7.70 │ 0.094133688738398 │\n", "│ 4 │ 3.70 │ 5.60 │ 0.125904416258132 │\n", "│ 5 │ 4.80 │ 6.80 │ 0.175203494390092 │\n", "│ 6 │ 4.90 │ 7.30 │ 0.136247013199614 │\n", "│ 7 │ 5.20 │ 8.30 │ 0.096520126235555 │\n", "│ 8 │ 3.60 │ 7.00 │ 0.035976773199740 │\n", "│ 9 │ 5.50 │ 7.40 │ 0.226615588095287 │\n", "│ 10 │ 4.20 │ 6.10 │ 0.154713629597006 │\n", "└─────────┴────────┴──────┴───────────────────┘\n" ] } ], "source": [ "function getP5(i)\n", " P=mkP0(mkA(Ls[i],Ms[i]))\n", " # remark the P[n] is stored off by one\n", " return sum(P[n+1] for n=5:27)\n", "end\n", "P5s=getP5.(Is)\n", "T5=[Ps Ls Ms P5s]\n", "pretty_table(T5,\n", " formatters=[fmt__printf(\"%3d\",[1]),\n", " fmt__printf(\"%.2f\",[2,3]),fmt__printf(\"%.15f\",[4])],\n", " column_labels=[\"version\",\"lambda\",\"mu\",\"P(Xt>=5)\"])" ] }, { "cell_type": "markdown", "id": "1117b2e8-9b35-4afe-bae8-0fb2bbcdde30", "metadata": {}, "source": [ "(iv) Now suppose that a second mechanic is\n", "called in whenever two or\n", "more forklifts need repair. State the new rate matrix $\\tilde A$ and find\n", "the corresponding steady-state distribution ${\\bf P}(\\tilde X_t=i)$\n", "of the number of forklifts in the facility." ] }, { "cell_type": "markdown", "id": "34efd466-5a7e-45eb-96d1-d19b05bd774a", "metadata": {}, "source": [ "The new rate diagram looks like" ] }, { "attachments": { "f596ffbc-be44-445a-beb3-18f057c276ab.png": { "image/png": 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HYHV19Q3j+9raWvU4v6amBsXFxcjPz0dhYaF6/AQA5ubmsLW1ha2tLbp3744ePXqgR48e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} }, "cell_type": "markdown", "id": "98a3f9e9-d108-4820-ae10-c0681ffa90b9", "metadata": {}, "source": [ "![forkrate2.png](attachment:f596ffbc-be44-445a-beb3-18f057c276ab.png)" ] }, { "cell_type": "markdown", "id": "899131bb-1cff-473b-9c1e-6f4109e997e4", "metadata": {}, "source": [ "Therefore, the corresponding matrix is\n", "$$\n", " \\tilde A=\\left[\\matrix{ -\\lambda&\\lambda&&&&0\\cr\n", " \\mu & -\\mu-\\lambda&\\lambda&&\\cr\n", " &2\\mu& -2\\mu-\\lambda&\\ddots\\cr\n", " && 2\\mu & \\ddots&\\lambda&\\cr\n", " &&& \\ddots&-2\\mu-\\lambda&\\lambda\\cr\n", " 0&&&& 2\\mu & -2\\mu\\cr\n", " }\\right].\n", "$$" ] }, { "cell_type": "code", "execution_count": 15, "id": "a4445c47-9e9b-4f72-a4fb-f78998e66a9d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "mktA (generic function with 1 method)" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "function mktA(lambda,mu)\n", " a1=[mu,2*mu*ones(Int,26)...]\n", " a3=lambda*ones(Int,27)\n", " a2=[-lambda,-(mu+lambda),-(2*mu+lambda)ones(Int,25)...,-2*mu]\n", " return Tridiagonal(a1,a2,a3)\n", "end" ] }, { "cell_type": "code", "execution_count": 16, "id": "96127c23-fe64-4d18-bca4-5008dfeb63ef", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "28×28 Tridiagonal{Num, Vector{Num}}:\n", " -λ λ ⋅ ⋅ … ⋅ ⋅ ⋅ ⋅\n", " μ -λ - μ λ ⋅ ⋅ ⋅ ⋅ ⋅\n", " ⋅ \u001b[96m2\u001b[39mμ -λ - \u001b[96m2\u001b[39mμ λ ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ \u001b[96m2\u001b[39mμ -λ - \u001b[96m2\u001b[39mμ ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ \u001b[96m2\u001b[39mμ ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n", " ⋮ ⋱ ⋮ \n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ … ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ λ ⋅ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ -λ - \u001b[96m2\u001b[39mμ λ ⋅ ⋅\n", " ⋅ ⋅ ⋅ ⋅ … \u001b[96m2\u001b[39mμ -λ - \u001b[96m2\u001b[39mμ λ ⋅\n", " ⋅ ⋅ ⋅ ⋅ ⋅ \u001b[96m2\u001b[39mμ -λ - \u001b[96m2\u001b[39mμ λ\n", " ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ \u001b[96m2\u001b[39mμ \u001b[96m-2\u001b[39mμ" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# check it is correct using Symbolics\n", "using Symbolics\n", "@variables λ,μ\n", "mktA(λ,μ)" ] }, { "cell_type": "code", "execution_count": 17, "id": "ca0bd367-5b1c-4ff6-a27e-3dfcca3fcb6f", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "┌──────────┬───────────────────┬──────────┬───────────────────┐\n", "│\u001b[1m n \u001b[0m│\u001b[1m tP0(n) \u001b[0m│\u001b[1m n \u001b[0m│\u001b[1m tP0(n) \u001b[0m│\n", "├──────────┼───────────────────┼──────────┼───────────────────┤\n", "│ 0 │ 0.502538071066017 │ 14 │ 0.000000191108529 │\n", "│ 1 │ 0.332761695705876 │ 15 │ 0.000000063272418 │\n", "│ 2 │ 0.110171101956675 │ 16 │ 0.000000020948301 │\n", "│ 3 │ 0.036475567539710 │ 17 │ 0.000000006935586 │\n", "│ 4 │ 0.012076370334093 │ 18 │ 0.000000002296241 │\n", "│ 5 │ 0.003998257745747 │ 19 │ 0.000000000760242 │\n", "│ 6 │ 0.001323747496903 │ 20 │ 0.000000000251702 │\n", "│ 7 │ 0.000438267752353 │ 21 │ 0.000000000083334 │\n", "│ 8 │ 0.000145102161252 │ 22 │ 0.000000000027590 │\n", "│ 9 │ 0.000048040580414 │ 23 │ 0.000000000009135 │\n", "│ 10 │ 0.000015905327299 │ 24 │ 0.000000000003024 │\n", "│ 11 │ 0.000005265952957 │ 25 │ 0.000000000001002 │\n", "│ 12 │ 0.000001743457398 │ 26 │ 0.000000000000331 │\n", "│ 13 │ 0.000000577225760 │ 27 │ 0.000000000000110 │\n", "└──────────┴───────────────────┴──────────┴───────────────────┘\n" ] } ], "source": [ "tP0=mkP0(mktA(Ls[1],Ms[1]))\n", "T6=[0:13 tP0[1:14] 14:27 tP0[15:28]]\n", "pretty_table(T6,\n", " formatters=[fmt__printf(\"%8d\",[1,3]),fmt__printf(\"%.15f\",[2,4])],\n", " column_labels=[\"n\",\"tP0(n)\",\"n\",\"tP0(n)\"])" ] }, { "cell_type": "markdown", "id": "37781857-bfa7-4a33-a0c4-f4ae1b46c922", "metadata": {}, "source": [ "Remark the above table is only for version 1." ] }, { "cell_type": "markdown", "id": "d3953314-ae67-472b-98aa-f3858055c27e", "metadata": {}, "source": [ "(vi) Use the results of part (v) to calculate the expected number\n", "${\\bf E}[\\tilde X_t]$ of\n", "forklifts in the repair facility, the probability that the first\n", "mechanic is busy, the probability that the second mechanic is called\n", "in and the probability that 5 or more forklifts are in the facility." ] }, { "cell_type": "markdown", "id": "0c76a46c-9d24-4e0d-bd71-704951ab16c8", "metadata": {}, "source": [ "For large $t$ the expected number of forklifts in the\n", "facility is given by\n", "$$\n", " {\\bf E}[\\tilde X_t]\\approx \\sum_{n=0}^{27} n\\tilde P_n.\n", "$$\n", "The probability that the first mechanic is busy is\n", "given by\n", "$$\n", " {\\bf P}(\\tilde X_t\\ge 1)=1-{\\bf P}(\\tilde X_t=0)\\approx 1-\\tilde P_0\n", "$$\n", "The probability that the second mechanic is called in is given by\n", "$$\n", " {\\bf P}(\\tilde X_t\\ge 2)=1-{\\bf P}(\\tilde X_t<2)\\approx 1-\\tilde P_0-\\tilde P_1\n", "$$\n", "and the probability that 5 or more forklifts are in the facility is\n", "$$\n", " {\\bf P}(\\tilde X_t\\ge 5)\\approx \\sum_{n=5}^{27} \\tilde P_n.\n", "$$" ] }, { "cell_type": "code", "execution_count": 18, "id": "f27ca469-f2f3-42ab-84a5-a1d286f3e581", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "┌─────────┬────────┬──────┬──────────┬───────────┬───────────┬───────────┐\n", "│\u001b[1m version \u001b[0m│\u001b[1m lambda \u001b[0m│\u001b[1m mu \u001b[0m│\u001b[1m E[tXt] \u001b[0m│\u001b[1m P(tXt>=1) \u001b[0m│\u001b[1m P(tXt>=2) \u001b[0m│\u001b[1m P(tXt>=5) \u001b[0m│\n", "├─────────┼────────┼──────┼──────────┼───────────┼───────────┼───────────┤\n", "│ 1 │ 4.90 │ 7.40 │ 0.743680 │ 0.497462 │ 0.164700 │ 0.005977 │\n", "│ 2 │ 4.80 │ 7.70 │ 0.690454 │ 0.475248 │ 0.148129 │ 0.004485 │\n", "│ 4 │ 3.70 │ 5.60 │ 0.741655 │ 0.496644 │ 0.164070 │ 0.005915 │\n", "│ 5 │ 4.80 │ 6.80 │ 0.806324 │ 0.521739 │ 0.184143 │ 0.008096 │\n", "│ 6 │ 4.90 │ 7.30 │ 0.756437 │ 0.502564 │ 0.168669 │ 0.006376 │\n", "│ 7 │ 5.20 │ 8.30 │ 0.694672 │ 0.477064 │ 0.149442 │ 0.004594 │\n", "│ 8 │ 3.60 │ 7.00 │ 0.550699 │ 0.409091 │ 0.105195 │ 0.001789 │\n", "│ 9 │ 5.50 │ 7.40 │ 0.862334 │ 0.541872 │ 0.201371 │ 0.010335 │\n", "│ 10 │ 4.20 │ 6.10 │ 0.781098 │ 0.512195 │ 0.176329 │ 0.007194 │\n", "└─────────┴────────┴──────┴──────────┴───────────┴───────────┴───────────┘\n" ] } ], "source": [ "function gettE(i)\n", " tP=mkP0(mktA(Ls[i],Ms[i]))\n", " # remark the P[n] is stored off by one\n", " return sum(n*tP[n+1] for n=0:27)\n", "end\n", "tEs=gettE.(Is)\n", "tP0s=[mkP0(mktA(Ls[i],Ms[i]))[1] for i=Is]\n", "tP2s=[1-sum(mkP0(mktA(Ls[i],Ms[i]))[1:2]) for i=Is]\n", "tP5s=[sum(mkP0(mktA(Ls[i],Ms[i]))[6:28]) for i=Is]\n", "T7=[Ps Ls Ms tEs 1.0.-tP0s tP2s tP5s]\n", "pretty_table(T7,\n", " formatters=[fmt__printf(\"%3d\",[1]),\n", " fmt__printf(\"%.2f\",[2,3]),\n", " fmt__printf(\"%.6f\",[4,5,6,7])],\n", " column_labels=[\"version\",\"lambda\",\"mu\",\"E[tXt]\",\n", " \"P(tXt>=1)\",\"P(tXt>=2)\",\"P(tXt>=5)\"])" ] }, { "cell_type": "markdown", "id": "43c5bb0d-aee9-47c2-8f7e-c3a43ea5cddf", "metadata": {}, "source": [ "(vii) Compare the results in (vi) to the results with only one mechanic." ] }, { "cell_type": "code", "execution_count": 19, "id": "6784ea78-4721-4503-b3fe-86ffdba1e884", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "9×7 Matrix{Any}:\n", " 1 4.9 7.4 [1.95973 0.74368] … [0.12729 0.00597719]\n", " 2 4.8 7.7 [1.65512 0.690454] [0.0941337 0.00448541]\n", " 4 3.7 5.6 [1.94711 0.741655] [0.125904 0.00591535]\n", " 5 4.8 6.8 [2.39837 0.806324] [0.175203 0.00809586]\n", " 6 4.9 7.3 [2.04127 0.756437] [0.136247 0.00637624]\n", " 7 5.2 8.3 [1.67736 0.694672] … [0.0965201 0.00459365]\n", " 8 3.6 7.0 [1.05882 0.550699] [0.0359768 0.00178862]\n", " 9 5.5 7.4 [2.88784 0.862334] [0.226616 0.0103348]\n", " 10 4.2 6.1 [2.20972 0.781098] [0.154714 0.00719438]" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P2s=[1-sum(mkP0(mkA(Ls[i],Ms[i]))[1:2]) for i=Is]\n", "T8t=[[Ps[i],Ls[i],Ms[i],[Es[i],tEs[i]],\n", " [1-P0s[i],1-tP0s[i]],[P2s[i],tP2s[i]],[P5s[i],tP5s[i]]]\n", " for i=Is]\n", "T8=reduce(vcat,T8t')" ] }, { "cell_type": "code", "execution_count": 20, "id": "daf064fc-669c-4f79-b1c1-04068c09b955", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "┌─────────┬────────┬──────┬──────────┬──────────┬──────────┬──────────┐\n", "│\u001b[1m version \u001b[0m│\u001b[1m lambda \u001b[0m│\u001b[1m mu \u001b[0m│\u001b[1m E[Xt] \u001b[0m│\u001b[1m P(Xt>=1) \u001b[0m│\u001b[1m P(Xt>=2) \u001b[0m│\u001b[1m P(Xt>=5) \u001b[0m│\n", "├─────────┼────────┼──────┼──────────┼──────────┼──────────┼──────────┤\n", "│ 1 │ 4.90 │ 7.40 │ 1.959728 │ 0.662159 │ 0.438453 │ 0.127290 │\n", "│ │ │ │ 0.743680 │ 0.497462 │ 0.164700 │ 0.005977 │\n", "├─────────┼────────┼──────┼──────────┼──────────┼──────────┼──────────┤\n", "│ 2 │ 4.80 │ 7.70 │ 1.655122 │ 0.623376 │ 0.388597 │ 0.094134 │\n", "│ │ │ │ 0.690454 │ 0.475248 │ 0.148129 │ 0.004485 │\n", "├─────────┼────────┼──────┼──────────┼──────────┼──────────┼──────────┤\n", "│ 4 │ 3.70 │ 5.60 │ 1.947113 │ 0.660711 │ 0.436538 │ 0.125904 │\n", "│ │ │ │ 0.741655 │ 0.496644 │ 0.164070 │ 0.005915 │\n", "├─────────┼────────┼──────┼──────────┼──────────┼──────────┼──────────┤\n", "│ 5 │ 4.80 │ 6.80 │ 2.398372 │ 0.705865 │ 0.498241 │ 0.175203 │\n", "│ │ │ │ 0.806324 │ 0.521739 │ 0.184143 │ 0.008096 │\n", "├─────────┼────────┼──────┼──────────┼──────────┼──────────┼──────────┤\n", "│ 6 │ 4.90 │ 7.30 │ 2.041269 │ 0.671228 │ 0.450546 │ 0.136247 │\n", "│ │ │ │ 0.756437 │ 0.502564 │ 0.168669 │ 0.006376 │\n", "├─────────┼────────┼──────┼──────────┼──────────┼──────────┼──────────┤\n", "│ 7 │ 5.20 │ 8.30 │ 1.677362 │ 0.626505 │ 0.392509 │ 0.096520 │\n", "│ │ │ │ 0.694672 │ 0.477064 │ 0.149442 │ 0.004594 │\n", "├─────────┼────────┼──────┼──────────┼──────────┼──────────┼──────────┤\n", "│ 8 │ 3.60 │ 7.00 │ 1.058823 │ 0.514286 │ 0.264490 │ 0.035977 │\n", "│ │ │ │ 0.550699 │ 0.409091 │ 0.105195 │ 0.001789 │\n", "├─────────┼────────┼──────┼──────────┼──────────┼──────────┼──────────┤\n", "│ 9 │ 5.50 │ 7.40 │ 2.887836 │ 0.743180 │ 0.552300 │ 0.226616 │\n", "│ │ │ │ 0.862334 │ 0.541872 │ 0.201371 │ 0.010335 │\n", "├─────────┼────────┼──────┼──────────┼──────────┼──────────┼──────────┤\n", "│ 10 │ 4.20 │ 6.10 │ 2.209716 │ 0.688516 │ 0.474051 │ 0.154714 │\n", "│ │ │ │ 0.781098 │ 0.512195 │ 0.176329 │ 0.007194 │\n", "└─────────┴────────┴──────┴──────────┴──────────┴──────────┴──────────┘\n" ] } ], "source": [ "using Printf\n", "printcell(v,_,_)=@sprintf(\"%.6f\\n%.6f\",v[1],v[2])\n", "pretty_table(T8,\n", " line_breaks=true,\n", " table_format=TextTableFormat(horizontal_lines_at_data_rows=:all),\n", " formatters=[fmt__printf(\"%3d\",[1]),\n", " fmt__printf(\"%.2f\",[2,3]),(v,i,j)->j>3 ? printcell(v,i,j) : v],\n", " column_labels=[\"version\",\"lambda\",\"mu\",\"E[Xt]\",\n", " \"P(Xt>=1)\",\"P(Xt>=2)\",\"P(Xt>=5)\"],\n", " fit_table_in_display_vertically=false)" ] }, { "cell_type": "markdown", "id": "b5229f4f-7e46-46c2-b09c-2f91985535bd", "metadata": {}, "source": [ "For each multi-line entry of the table of the form\n", "```\n", "┼──────────┼\n", "│ 1.959728 │\n", "│ 0.720465 │\n", "┼──────────┼\n", "```\n", "the first line indicates the original statistic with only\n", "one mechanic available while the second line indicates\n", "the new statistic when the second mechanic is available.\n", "\n", "For all project versions, adding a second mechanic reduces the expected repair\n", "queue length, decreases the percentage of time the first mechanic is\n", "working and decreases the percentage of time there are five or more\n", "forklifts waiting in the queue." ] }, { "cell_type": "markdown", "id": "61ef5db1-bc12-4ea6-9121-91d9ae9ae13b", "metadata": {}, "source": [ "(viii) The cost of the second mechanic is \\\\$250 per day and the mechanic \n", "is only paid for the days worked. Another option is to lease replacement\n", "forklifts for the customers whose forklifts are in for repair but only\n", "during periods of backlog when two or more forklifts are in the facility.\n", "If the cost of replacements is \\\\$25 per day per forklift,\n", "which of the two plans is most cost effective?" ] }, { "cell_type": "markdown", "id": "3c8e75f5-ecde-44d5-a5e7-b3d79c26b079", "metadata": {}, "source": [ "Since a second mechanic only works when there is backlog, the expected cost \n", "of the second mechanic is\n", "$$\n", " C_1=250{\\bf P}(\\tilde X_t\\ge 2)\\approx 250 \\tilde P_2\n", "$$" ] }, { "cell_type": "code", "execution_count": 21, "id": "b02c477a-b2aa-4e96-801b-af1e102896b5", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "41.17505830702667" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Expected daily cost for second mechanic\n", "C1(i)=250*tP2s[i]\n", "C1(1)" ] }, { "cell_type": "markdown", "id": "fb24c0db-4e48-415a-b959-ffcdd40817e8", "metadata": {}, "source": [ "For leasing forklifts I assumed\n", " \n", "- All customers whose forklifts are in the\n", " shop get a replacement forklift during times of\n", " backlog, including the forklift being repaired.\n", "\n", "Note that it may have happened that the first forklift came into the shop\n", "at a time there was no backlog, so the\n", "mechanic immediately started repair and no replacement forklift provided.\n", "However, before the repair finished another forklift could\n", "break down. In this case repairs continue on the first forklift, but\n", "the first customer is now entitled to a replacement forklift.\n", "Curiously, this\n", "would encourage customers to bring in their broken forklifts as batches.\n", "\n", "On the other hand, it may have happened that the\n", "forklift currently being repaired came into the shop when there was backlog\n", "but now that it's being repared there is no backlog. The moment repairs start,\n", "the previously obtained replacment forklift has to be returned. \n", "In my opinion it would\n", "seem unfair a customer has to return the replacement forklift before getting\n", "their own back from the shop." ] }, { "cell_type": "markdown", "id": "de65d1e1-3c83-442a-ba99-3aea3d011ed2", "metadata": {}, "source": [ "Our first goal is to compute the expected value of the\n", "queue length under the assumption of backlog.\n", "$$\n", " {\\bf E}[X_t | X_t\\ge 2]\n", " =\\sum_{n=2}^{27} n P(X_t=n | X_t\\ge 2)\n", " =\\sum_{n=2}^{27} {n P(X_t=n)\n", " \\over P(X_t\\ge 2)}.\n", "$$\n", "The expected cost is then\n", "$$\n", " C_2=25{\\bf E}[X_t | X_t\\ge 2] P(X_t\\ge 2)\n", " \\approx\\sum_{n=2}^{27} 25n P_n=25({\\bf E}[X_t]-P_1).\n", "$$" ] }, { "cell_type": "code", "execution_count": 22, "id": "ef9471a3-e2bd-410e-990f-110ec559aa6c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "43.40056622516525" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Expected daily cost for leased forklifts\n", "C2(i)=25*(Es[i]-P0[2])\n", "C2(1)" ] }, { "cell_type": "code", "execution_count": 23, "id": "f15125dc-27e6-45be-b918-a7bbe5a08757", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "┌─────────┬────────┬─────┬──────────┬───────────┬───────────┐\n", "│\u001b[1m version \u001b[0m│\u001b[1m lambda \u001b[0m│\u001b[1m mu \u001b[0m│\u001b[1m second \u001b[0m│\u001b[1m leased \u001b[0m│\u001b[1m recommend \u001b[0m│\n", "│\u001b[1m \u001b[0m│\u001b[1m \u001b[0m│\u001b[1m \u001b[0m│\u001b[1m mechanic \u001b[0m│\u001b[1m forklifts \u001b[0m│\u001b[1m \u001b[0m│\n", "├─────────┼────────┼─────┼──────────┼───────────┼───────────┤\n", "│ 1 │ 4.9 │ 7.4 │ 41.18 │ 43.40 │ mechanic │\n", "│ 2 │ 4.8 │ 7.7 │ 37.03 │ 35.79 │ lease │\n", "│ 4 │ 3.7 │ 5.6 │ 41.02 │ 43.09 │ mechanic │\n", "│ 5 │ 4.8 │ 6.8 │ 46.04 │ 54.37 │ mechanic │\n", "│ 6 │ 4.9 │ 7.3 │ 42.17 │ 45.44 │ mechanic │\n", "│ 7 │ 5.2 │ 8.3 │ 37.36 │ 36.34 │ lease │\n", "│ 8 │ 3.6 │ 7.0 │ 26.30 │ 20.88 │ lease │\n", "│ 9 │ 5.5 │ 7.4 │ 50.34 │ 66.60 │ mechanic │\n", "│ 10 │ 4.2 │ 6.1 │ 44.08 │ 49.65 │ mechanic │\n", "└─────────┴────────┴─────┴──────────┴───────────┴───────────┘\n" ] } ], "source": [ "C1s=C1.(Is)\n", "C2s=C2.(Is)\n", "Rs=[C1s[i]