{ "cells": [ { "cell_type": "markdown", "id": "e09a56c4-e88c-48b5-9696-52669e00223f", "metadata": {}, "source": [ "2. A pig weighing 200 pounds gains 5 pounds per day and costs\n", "45 cents a day to keep.\n", "Suppose the\n", "price for pigs after $t$ days is\n", "$$p(t) = 0.65e^{-(.01/.65)t}\\hbox{ dollars/pound}.$$" ] }, { "cell_type": "markdown", "id": "a652a038-418a-4b7a-bacd-5d7916a9875c", "metadata": {}, "source": [ "(i) Show the price for pigs is falling by one cent/day at time $t = 0$.\n", "What happens as $t$ increases?" ] }, { "cell_type": "markdown", "id": "b211b78d-416c-428a-97ba-c4533c952b4c", "metadata": {}, "source": [ "Differentiating as\n", "$$\n", " p'(t)=0.65e^{-(.01/.65)t}\\Big({.01\\over.64}\\Big)=-.01e^{-(.01/.65)t}\n", "$$\n", "and evaluating at $t=0$ as\n", "$$\n", " p'(0)=-.01e^{-(.01/.65)(0)}=-.01e^0=-.01 \n", "$$\n", "shows the price is falling by one cent/day." ] }, { "cell_type": "markdown", "id": "44c9de85-c044-4260-9e56-67df5ddd52de", "metadata": {}, "source": [ "As $t$ increases the value of the pig decreases exponentially.\n", "This means the amount of decrease is less over time. Note,\n", "unlike the linear approximation, the price per pound of the\n", "pig never goes negative.\n", "\n", "Graphically the price is" ] }, { "cell_type": "code", "execution_count": 1, "id": "1bd7b71c-a934-4173-8866-d36d2645b710", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "1-element Vector{Num}:\n", " t" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "using Symbolics\n", "D(f,x)=expand_derivatives(Differential(x)(f))\n", "@variables t" ] }, { "cell_type": "code", "execution_count": 2, "id": "72f17874-4121-45b0-b966-6e9554c3a5f1", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\u001b[96m0.65\u001b[39mexp(\u001b[96m-0.015384615384615384\u001b[39mt)" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "p(t)=0.65*exp(-(.01/.65)*t)\n", "p(t)" ] }, { "cell_type": "code", "execution_count": 3, "id": "8bcf3c92-e77e-4bf0-95c5-028e16710078", "metadata": {}, "outputs": [], "source": [ "using Plots" ] }, { "cell_type": "code", "execution_count": 4, "id": "dd0f9372-a7c9-481b-a254-0ed2ae763cd5", "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" ], "text/html": [ "" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot(p,0:0.1:100,\n", " xlabel=\"t\",ylabel=\"p(t)\",title=\"Dollars Per Pound vs Time\",\n", " legend=false)" ] }, { "cell_type": "markdown", "id": "3ad42c76-900d-487f-ac5c-d8c839bdcd9e", "metadata": {}, "source": [ "(ii) Plot the profit $P(t)$ versus the time $t$ to sell the pig." ] }, { "cell_type": "code", "execution_count": 5, "id": "b5bb3ad7-55db-4d37-8ae0-2646eabb14c5", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "P (generic function with 1 method)" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "w(t)=200+5*t\n", "C(t)=0.45*t\n", "R(t)=w(t)*p(t)\n", "P(t)=R(t)-C(t)" ] }, { "cell_type": "code", "execution_count": 6, "id": "4740f571-da0c-4e60-afde-8a879471344b", "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" ], "text/html": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "plot(P,0:0.1:25,\n", " xlabel=\"t\",ylabel=\"P(t)\",title=\"Profit vs Time\",\n", " legend=false)" ] }, { "cell_type": "markdown", "id": "e9afadd2-5f35-4a48-9f39-8f35b8e872de", "metadata": {}, "source": [ "Visual inspection of the plot suggests the optimal time\n", "to sell the pig is arount $t\\approx 14$." ] }, { "cell_type": "markdown", "id": "81993c2a-db79-47ae-a3f8-7ab3f9ab049a", "metadata": {}, "source": [ "(iii) Find the optimal time to sell the pig using Newton's method.\n", "Model as a one-variable non-linear optimization problem." ] }, { "cell_type": "code", "execution_count": 7, "id": "704421e9-eb6c-4750-bd34-c71beaef68c6", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-\u001b[96m0.45\u001b[39m + \u001b[96m1.25\u001b[39mexp(\u001b[96m-0.015384615384615384\u001b[39mt) - \u001b[96m0.05\u001b[39mt*exp(\u001b[96m-0.015384615384615384\u001b[39mt)" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dpn=expand(D(P(t),t))\n", "dps=\"dp(t)=\"*string(Symbolics.toexpr(dpn))\n", "eval(Meta.parse(dps))\n", "dp(t)" ] }, { "cell_type": "markdown", "id": "81e625a6-d5c7-45e4-bb3a-d8c14bd20a22", "metadata": {}, "source": [ "To find the maximum of of $P(t)$ find $t$ such that $P'(t)=0$.\n", "Do this by writing $g(t)=P'(t)$ and then using Newton's method\n", "$$\n", " t_{n+1}=t_n-g(t_n)/g'(t_n)\n", " \\qquad\\hbox{where}\\qquad\n", " t_0=14.\n", "$$\n", "Note the value $t_0=14$ was chosen because that is close to\n", "the observed maximum in the graph." ] }, { "cell_type": "code", "execution_count": 8, "id": "6d28b8f3-a5ce-4c64-aa28-0cb9771dafb7", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-\u001b[96m0.06923076923076923\u001b[39mexp(\u001b[96m-0.015384615384615384\u001b[39mt) + \u001b[96m0.0007692307692307692\u001b[39mt*exp(\u001b[96m-0.015384615384615384\u001b[39mt)" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(t)=dp(t)\n", "dgn=expand(D(g(t),t))\n", "dgs=\"dg(t)=\"*string(Symbolics.toexpr(dgn))\n", "eval(Meta.parse(dgs))\n", "dg(t)" ] }, { "cell_type": "code", "execution_count": 9, "id": "6b24dcdc-0e06-42b0-a016-326cc8ff5cca", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1 13.860549574911676\n", "2 13.860826286794973\n", "3 13.860826287886802\n", "4 13.860826287886802\n", "\n", "topt=13.860826287886802\n", "Popt=135.19404935454668\n" ] } ], "source": [ "tn=14\n", "for n=1:4\n", " tn=tn-g(tn)/dg(tn)\n", " println(n,\" \",tn)\n", "end\n", "topt=tn\n", "Popt=P(topt)\n", "println(\"\\ntopt=\",topt)\n", "println(\"Popt=\",Popt)" ] }, { "cell_type": "markdown", "id": "79736fe4-3276-4feb-a030-d3cdbf440d11", "metadata": {}, "source": [ "The optimal time to sell the pig is $t_{\\rm opt}\\approx 13.86$ and the\n", "optimal profit is $P_{\\rm opt}\\approx 135.19$." ] }, { "cell_type": "markdown", "id": "ac7f9f9e-ef19-44b9-8981-79a97cb758b6", "metadata": {}, "source": [ "(iv) The parameter 0.01 represents the rate $\\alpha$ at which price\n", "is falling at $t = 0$. Thus,\n", "$$p(t) = 0.65e^{-(\\alpha/.65)t}\\hbox{ dollars/pound}.$$\n", "Perform a sensitivity analysis with respect to the parameter $\\alpha$.\n", "Compute $S(t,\\alpha)$ and $S(P,\\alpha)$ evaluated at $\\alpha=0.01$.\n", "Here $P$ is the profit obtained at the optimal time $t$ to sell the pig." ] }, { "cell_type": "markdown", "id": "618ec500-444b-406a-9b18-14e4a07b0431", "metadata": {}, "source": [ "Since $g(t_{\\rm opt})=0$ use implicit differentiation\n", "to compute $dt/d\\alpha$ as follows\n", "$$\n", " {dg\\over d\\alpha} = {\\partial g\\over\\partial t}{dt\\over d\\alpha} + {\\partial g\\over\\partial \\alpha} = 0.\n", "$$\n", "It follows that\n", "$$\n", " {dt\\over d\\alpha}\\bigg|_{t=t_{\\rm opt}}=-{\\partial g\\over\\partial \\alpha}\\Big / {\\partial g\\over\\partial t}\n", " \\bigg|_{t=t_{\\rm opt}}.\n", "$$" ] }, { "cell_type": "code", "execution_count": 10, "id": "425c9ba3-a94c-4392-8393-3060858c174c", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\u001b[96m0.65\u001b[39mexp(\u001b[96m-1.5384615384615383\u001b[39malpha*t)" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "@variables alpha\n", "p(t)=0.65*exp(-(alpha/.65)*t)\n", "p(t)" ] }, { "cell_type": "code", "execution_count": 11, "id": "f806369f-63f3-40de-83ae-d644b7675a62", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-\u001b[96m0.45\u001b[39mt + \u001b[96m0.65\u001b[39m(\u001b[96m200\u001b[39m + \u001b[96m5\u001b[39mt)*exp(\u001b[96m-1.5384615384615383\u001b[39malpha*t)" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P(t)" ] }, { "cell_type": "code", "execution_count": 12, "id": "d7c968ce-7bcd-4a9e-84c9-50f54077e500", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-\u001b[96m0.45\u001b[39m + \u001b[96m3.25\u001b[39mexp(\u001b[96m-1.5384615384615383\u001b[39malpha*t) - \u001b[96m200.0\u001b[39malpha*exp(\u001b[96m-1.5384615384615383\u001b[39malpha*t) - \u001b[96m5.0\u001b[39malpha*t*exp(\u001b[96m-1.5384615384615383\u001b[39malpha*t)" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dpn=expand(D(P(t),t))\n", "dps=\"dp(t)=\"*string(Symbolics.toexpr(dpn))\n", "eval(Meta.parse(dps))\n", "dp(t)" ] }, { "cell_type": "code", "execution_count": 13, "id": "47bbc7e7-8ae0-4d64-bfea-92df533e2e95", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-\u001b[96m200.0\u001b[39mexp(\u001b[96m-1.5384615384615383\u001b[39malpha*t) - \u001b[96m10.0\u001b[39mt*exp(\u001b[96m-1.5384615384615383\u001b[39malpha*t) + \u001b[96m307.6923076923077\u001b[39malpha*t*exp(\u001b[96m-1.5384615384615383\u001b[39malpha*t) + \u001b[96m7.692307692307692\u001b[39malpha*(t^\u001b[96m2\u001b[39m)*exp(\u001b[96m-1.5384615384615383\u001b[39malpha*t)" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(t)=dp(t)\n", "pgpalpha=D(g(t),alpha)" ] }, { "cell_type": "code", "execution_count": 14, "id": "fdcbcd33-9f7f-4c1d-970d-8e322556253b", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-\u001b[96m10.0\u001b[39malpha*exp(\u001b[96m-1.5384615384615383\u001b[39malpha*t) + \u001b[96m307.6923076923077\u001b[39m(alpha^\u001b[96m2\u001b[39m)*exp(\u001b[96m-1.5384615384615383\u001b[39malpha*t) + \u001b[96m7.692307692307692\u001b[39m(alpha^\u001b[96m2\u001b[39m)*t*exp(\u001b[96m-1.5384615384615383\u001b[39malpha*t)" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "pgpt=D(g(t),t)" ] }, { "cell_type": "code", "execution_count": 15, "id": "0c256fdb-fa3a-463d-baac-9052c6b5aeb9", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-4800.881966946403" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "vopt=[t=>topt,alpha=>0.01]\n", "dtdalpha=substitute(-pgpalpha/pgpt,vopt)" ] }, { "cell_type": "code", "execution_count": 16, "id": "97a7b0c4-b7f5-4a69-b67d-95229f903bdc", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "S(t,alpha)=-3.4636333124973726\n" ] } ], "source": [ "Stalpha=0.01/topt*dtdalpha\n", "println(\"S(t,alpha)=\",Stalpha)" ] }, { "cell_type": "markdown", "id": "21e946e8-dc61-47f8-a38e-acc3b6eaab9a", "metadata": {}, "source": [ "To compute $dP/d\\alpha$ at $t=t_{\\rm opt}$ note $P'(t_{\\rm opt})=0$ implies\n", "$$\n", " {dP\\over d\\alpha}\n", " ={\\partial P\\over\\partial t} {d\\alpha\\over d\\alpha}+{\\partial P\\over\\partial \\alpha}\n", " ={\\partial P\\over\\partial \\alpha}\\bigg|_{t=t_{\\rm opt}}.\n", "$$" ] }, { "cell_type": "code", "execution_count": 17, "id": "f0ff5556-7da6-47bf-a262-a1f574edb55d", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\u001b[96m-3732.7777844922293\u001b[39mexp(\u001b[96m-0.21324348135210464\u001b[39m)" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dPdalphan=substitute(D(P(t),alpha),vopt)" ] }, { "cell_type": "code", "execution_count": 18, "id": "cc1ab5df-ffb8-4718-b6e2-70b3b4336755", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-3015.9328625872377" ] }, "execution_count": 18, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Evaluate the expression\n", "dPdalpha=eval(Symbolics.toexpr(dPdalphan))" ] }, { "cell_type": "code", "execution_count": 19, "id": "272e3db2-eb36-47fd-b60b-9bc82edba433", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "S(P,alpha)=-0.22308177593511883\n" ] } ], "source": [ "SPalpha=0.01/Popt*dPdalpha\n", "println(\"S(P,alpha)=\",SPalpha)" ] }, { "cell_type": "markdown", "id": "1be2b705-fb4f-4c4b-af45-837ce85437a1", "metadata": {}, "source": [ "In summary, the sensitivities are\n", "$$\n", " S(t,\\alpha)\\approx -3.46\n", " \\qquad\\hbox{and}\\qquad\n", " S(P,\\alpha)\\approx -0.223.\n", "$$" ] }, { "cell_type": "markdown", "id": "26139305-3ffe-4a65-9c8c-f8390f7c5d17", "metadata": {}, "source": [ "Estimate using the difference quotients\n", "$$\n", " {dt\\over d\\alpha}\\approx {\\Delta t\\over\\Delta\\alpha}\n", " \\qquad\\hbox{and}\\qquad\n", " {dP\\over d\\alpha}\\approx {\\Delta P\\over\\Delta\\alpha}\n", "$$\n", "to confirm the above sensitivities." ] }, { "cell_type": "code", "execution_count": 20, "id": "2878314d-dc09-4ab1-ad4c-f2236deedf47", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\u001b[96m0.65\u001b[39mexp(\u001b[96m-0.015399999999999999\u001b[39mt)" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Take the perturbed alpha close to 0.01 for accuracy\n", "alphanew=0.01001\n", "alpha=alphanew\n", "p(t)" ] }, { "cell_type": "code", "execution_count": 21, "id": "5f56b9ba-db3e-4e7c-a642-0009f68c7983", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-\u001b[96m0.45\u001b[39m + \u001b[96m1.2480000000000002\u001b[39mexp(\u001b[96m-0.015399999999999999\u001b[39mt) - \u001b[96m0.05005\u001b[39mt*exp(\u001b[96m-0.015399999999999999\u001b[39mt)" ] }, "execution_count": 21, "metadata": {}, "output_type": "execute_result" } ], "source": [ "dpn=expand(D(P(t),t))\n", "dps=\"dp(t)=\"*string(Symbolics.toexpr(dpn))\n", "eval(Meta.parse(dps))\n", "dp(t)" ] }, { "cell_type": "code", "execution_count": 22, "id": "18ae2c0b-2ebc-4006-acd8-0aaa0055b790", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-\u001b[96m0.0692692\u001b[39mexp(\u001b[96m-0.015399999999999999\u001b[39mt) + \u001b[96m0.0007707699999999999\u001b[39mt*exp(\u001b[96m-0.015399999999999999\u001b[39mt)" ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "g(t)=dp(t)\n", "dgn=expand(D(g(t),t))\n", "dgs=\"dg(t)=\"*string(Symbolics.toexpr(dgn))\n", "eval(Meta.parse(dgs))\n", "dg(t)" ] }, { "cell_type": "code", "execution_count": 23, "id": "4f09c71b-c805-4b22-91d8-e6946d301463", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "1 13.812362974310679\n", "2 13.812864114635024\n", "3 13.812864118219844\n", "4 13.812864118219846\n", "\n", "tnew=13.812864118219846\n", "Pnew=135.1639477232832\n" ] } ], "source": [ "tn=14\n", "for n=1:4\n", " tn=tn-g(tn)/dg(tn)\n", " println(n,\" \",tn)\n", "end\n", "tnew=tn\n", "Pnew=P(tnew)\n", "println(\"\\ntnew=\",tnew)\n", "println(\"Pnew=\",Pnew)" ] }, { "cell_type": "code", "execution_count": 24, "id": "ccb484a0-12df-4314-98e6-235601b75a77", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-4796.216966695805" ] }, "execution_count": 24, "metadata": {}, "output_type": "execute_result" } ], "source": [ "deltat=(tnew-topt)/(alphanew-0.01)" ] }, { "cell_type": "code", "execution_count": 25, "id": "fa399ea1-4f1b-4f21-a1b2-f38781fa98f0", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "-3010.163126347843" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "deltaP=(Pnew-Popt)/(alphanew-0.01)" ] }, { "cell_type": "markdown", "id": "3196fd35-39ca-4421-9c5c-3ad043838ca0", "metadata": {}, "source": [ "The consistency between\n", "$$\n", " -4800.881\\approx {dt\\over d\\alpha}\\approx {\\Delta t\\over\\Delta\\alpha}\\approx -4796.22\n", " \\qquad\\hbox{and}\\qquad\n", " -3015.93\\approx {dP\\over d\\alpha}\\approx {\\Delta P\\over\\Delta\\alpha}\\approx -3010.16\n", "$$\n", "suggest the earlier computations for $S(t,\\alpha)$ and $S(P,\\alpha)$ are correct." ] }, { "cell_type": "code", "execution_count": null, "id": "8391bae8-2696-4b0c-8f4b-85e82a4c9a0e", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.10", "language": "julia", "name": "julia-1.10" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.10.10" } }, "nbformat": 4, "nbformat_minor": 5 }