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<language math="en" UI="en"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="false" optimize-expressions="false" exact-boolean="true" strings-use-origin="false" zero-over-zero="error"> <compatibility multiple-assignment="MC12" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="mks" customized="false"/> </units> </calculation> <editor view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"/> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="true" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="907" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="7536672"/> </miscellaneous> </settings> <regions> <region region-id="1" left="6" top="2.25" width="246.75" height="24" align-x="6" align-y="12" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Problem 9.2 of Shuler & Kargi.<sp count="2"/>Two fermentors in series. Instructor: Nam Sun Wang </text> </region> <region region-id="2" left="6" top="30" width="436.5" height="9.75" align-x="6" align-y="30" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <picture> <png item-idref="2" display-width="435" display-height="8.25"/> </picture> <rendering item-idref="3"/> </region> <region region-id="367" left="6" top="44.25" width="102.75" height="12" align-x="6" align-y="54" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Cell growth parameters: </text> </region> <region region-id="349" left="12" top="63" width="36.75" height="15" align-x="30" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id 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</region> <region region-id="884" left="126" top="56.25" width="18.75" height="27.75" align-x="133.5" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:div/> <ml:id xml:space="preserve">g</ml:id> <ml:id xml:space="preserve">liter</ml:id> </ml:apply> </math> <rendering item-idref="7"/> </region> <region region-id="347" left="156" top="54" width="64.5" height="32.25" align-x="183.75" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="1">μ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">s</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:apply> <ml:mult/> <ml:id 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z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="s">Y</ml:id> <ml:real>0.4</ml:real> </ml:define> </math> <rendering item-idref="10"/> </region> <region region-id="570" left="336" top="56.25" width="27.75" height="27.75" align-x="348" align-y="72" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:div/> <ml:id xml:space="preserve">g_cell</ml:id> <ml:id xml:space="preserve">g_S</ml:id> </ml:apply> </math> <rendering item-idref="11"/> </region> <region region-id="410" left="6" top="86.25" width="131.25" height="12" align-x="6" align-y="96" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Product formation parameters: </text> </region> <region region-id="403" left="12" top="105" width="34.5" height="15" align-x="27.75" align-y="114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="p">Y</ml:id> <ml:real>0.6</ml:real> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="409" left="54" top="98.25" width="19.5" height="27.75" align-x="62.25" align-y="114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:div/> <ml:id xml:space="preserve">g_P</ml:id> <ml:id xml:space="preserve">g_S</ml:id> </ml:apply> </math> <rendering item-idref="13"/> </region> <region region-id="405" left="90" top="105" width="38.25" height="15" align-x="105" align-y="114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="p">q</ml:id> <ml:real>0.02</ml:real> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="411" left="138" top="98.25" width="39" height="27.75" align-x="150.75" align-y="114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:div/> <ml:id xml:space="preserve">g_P</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">g_cell</ml:id> <ml:id xml:space="preserve">hr</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="15"/> </region> <region region-id="361" left="6" top="128.25" width="85.5" height="12" align-x="6" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Substrate feed rate: </text> </region> <region region-id="352" left="12" top="147" width="32.25" height="12.75" align-x="23.25" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">F</ml:id> <ml:real>100</ml:real> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="421" left="48" top="140.25" width="18.75" height="27.75" align-x="55.5" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:div/> <ml:id xml:space="preserve">liter</ml:id> <ml:id xml:space="preserve">hr</ml:id> </ml:apply> </math> <rendering item-idref="17"/> </region> <region region-id="359" left="84" top="147" width="24.75" height="15" align-x="96.75" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="f">s</ml:id> <ml:real>5</ml:real> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="422" left="114" top="140.25" width="18.75" height="27.75" align-x="121.5" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:div/> <ml:id xml:space="preserve">gm</ml:id> <ml:id xml:space="preserve">liter</ml:id> </ml:apply> </math> <rendering item-idref="19"/> </region> <region region-id="894" left="168" top="147" width="69" height="15" align-x="205.5" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="washout">D</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="1">μ</ml:id> <ml:id xml:space="preserve" subscript="f">s</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="876" left="6" top="170.25" width="195.75" height="12" align-x="6" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Steady-state equations for the first fermentor. </text> </region> <region region-id="877" left="216" top="171" width="36.75" height="15" align-x="231.75" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">V</ml:id> <ml:real>500</ml:real> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="878" left="258" top="171" width="15.75" height="12.75" align-x="264.75" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">liter</ml:id> </math> <rendering item-idref="22"/> </region> <region region-id="879" left="294" top="164.25" width="65.25" height="30" align-x="310.5" align-y="180" show-border="false" 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</ml:apply> </ml:apply> </math> <rendering item-idref="29"/> </region> <region region-id="426" left="6" top="273" width="90" height="15" align-x="42.75" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">xs1</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:Find auto-method="false" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve" subscript="1">x</ml:id> <ml:id xml:space="preserve" subscript="1">s</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="688" left="126" top="273" width="74.25" height="15" 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subscript="1">s</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:indexer/> <ml:apply> <ml:id xml:space="preserve">xs1</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="690" left="330" top="264.75" width="78.75" height="29.25" align-x="366" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">xs1</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:real>1.9199999999999995</ml:real> <ml:real>0.20000000000000015</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="33"/> </region> <region region-id="362" left="6" top="296.25" width="194.25" height="12" align-x="6" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Dynamic equations for the second fermentor. </text> </region> <region region-id="881" left="216" top="297" width="36.75" height="15" align-x="231.75" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">V</ml:id> <ml:real>300</ml:real> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="882" left="258" top="297" width="15.75" height="12.75" align-x="264.75" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">liter</ml:id> </math> <rendering item-idref="35"/> </region> <region region-id="883" left="294" top="290.25" width="74.25" height="30" align-x="310.5" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve">F</ml:id> <ml:id xml:space="preserve" subscript="2">V</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.33333333333333331</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="36"/> </region> <region region-id="892" left="372" top="297" width="21" height="18.75" align-x="381.75" align-y="312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:pow/> <ml:id xml:space="preserve">hr</ml:id> <ml:real>-1</ml:real> </ml:apply> </math> <rendering item-idref="37"/> </region> <region region-id="691" left="6" top="315" width="25.5" height="15" align-x="19.5" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">x</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="692" left="48" top="315" width="26.25" height="15" align-x="62.25" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="693" left="96" top="315" width="27" height="15" align-x="111" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">p</ml:id> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="40"/> </region> <region region-id="694" left="138" top="314.25" width="63.75" height="12" align-x="138" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> ... initial guess </text> </region> <region region-id="714" left="6" top="333" width="24" height="12.75" align-x="17.25" align-y="342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="41"/> </region> <region region-id="706" left="18" top="344.25" width="26.25" height="15" align-x="18" align-y="354" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> dx2/dt </text> </region> <region region-id="695" left="66" top="345" width="129.75" height="15" align-x="75.75" align-y="354" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="1">x</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="2">x</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve" subscript="2">μ</ml:id> <ml:id xml:space="preserve" subscript="2">s</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="2">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="42"/> </region> <region region-id="707" left="18" top="368.25" width="26.25" height="15" align-x="18" align-y="378" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> ds2/dt </text> </region> <region region-id="708" left="66" top="362.25" width="192" height="30" align-x="75.75" align-y="378" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="1">s</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="2">s</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="s">Y</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="2">μ</ml:id> <ml:id xml:space="preserve" subscript="2">s</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="2">x</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="p">Y</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="p">q</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="2">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="43"/> </region> <region region-id="713" left="18" top="392.25" width="26.25" height="15" align-x="18" align-y="402" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> ds2/dt </text> </region> <region region-id="711" left="66" top="393" width="69.75" height="15" align-x="75.75" align-y="402" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="p">q</ml:id> <ml:id xml:space="preserve" subscript="2">x</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:id xml:space="preserve" subscript="2">p</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="44"/> </region> <region region-id="715" left="12" top="411" width="123.75" height="15" align-x="69" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">xsp2</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:Find auto-method="false" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve" subscript="2">x</ml:id> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:id xml:space="preserve" subscript="2">p</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="45"/> </region> <region region-id="725" left="150" top="411" width="110.25" height="15" align-x="197.25" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:indexer/> <ml:apply> <ml:id xml:space="preserve">xsp2</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="726" left="282" top="411" width="111" height="15" align-x="330" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="2">p</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:indexer/> <ml:apply> <ml:id xml:space="preserve">xsp2</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="744" left="318" top="434.25" width="120" height="24" align-x="318" align-y="444" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> The value of s is 0.01X the value of p. </text> </region> <region region-id="718" left="12" top="432" width="110.25" height="47.25" align-x="69.75" align-y="450" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">prod</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve" subscript="2">p</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="48"/> </region> <region region-id="742" left="156" top="432" width="150" height="47.25" align-x="216.75" align-y="450" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">profit</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="2">p</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:mult/> <ml:real>0.01</ml:real> <ml:id xml:space="preserve" subscript="f">s</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="49"/> </region> <region region-id="740" left="156" top="495" width="90" height="15" align-x="213" align-y="504" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">prod</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.014399999999999998</ml:real> </result> </ml:eval> </math> <rendering item-idref="50"/> </region> <region region-id="741" left="258" top="481.5" width="120" height="51.75" align-x="314.25" align-y="510" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">xsp2</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="3" cols="1"> <ml:real>1.9199999999999995</ml:real> <ml:real>0.0080000000000001511</ml:real> <ml:real>0.11519999999999998</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="51"/> </region> <region region-id="895" left="0" top="534" width="6000" height="6" align-x="0" align-y="534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="745" left="6" top="537" width="72" height="15" align-x="22.5" align-y="546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:range> <ml:sequence> <ml:real>0.01</ml:real> <ml:real>0.02</ml:real> </ml:sequence> <ml:real>1</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="52"/> </region> <region region-id="746" left="6" top="552" width="255" height="144" align-x="6" align-y="552" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="53"/> <rendering item-idref="54"/> </region> <region region-id="747" left="264" top="566.25" width="169.5" height="69" align-x="264" align-y="576" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Product concentration and product productivity continue to rise as D2 <f family="Symbol" charset="2"></f>0.<sp count="2"/>However, this is only because we failed to impose the constraint 0<f family="Symbol" charset="2"></f>s2.<sp count="2"/>Notice 0<f family="Symbol" charset="2"></f>s2 for small values of D2. </text> </region> <region region-id="777" left="6" top="698.25" width="456.75" height="15" align-x="6" align-y="708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Optimization without constraint <f family="Symbol" charset="2">  </f> high product productivity, but unreal because of negative D 2 . </text> </region> <region region-id="780" left="12" top="723" width="35.25" height="15" align-x="28.5" align-y="732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="55"/> </region> <region region-id="781" left="60" top="723" width="35.25" height="15" align-x="76.5" align-y="732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:real>0.3</ml:real> </ml:define> </math> <rendering item-idref="56"/> </region> <region region-id="782" left="120" top="712.5" width="188.25" height="33.75" align-x="150.75" align-y="732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:matrix> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:Maximize auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve">prod</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:real>0.048625339429016104</ml:real> <ml:real>-0.0486252671320941</ml:real> </ml:matrix> </result> </ml:eval> </ml:define> </math> <rendering item-idref="57"/> </region> <region region-id="783" left="324" top="717" width="110.25" height="21" align-x="381" align-y="732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">prod</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>26799.049583556956</ml:real> </result> </ml:eval> </math> <rendering item-idref="58"/> </region> <region region-id="785" left="120" top="748.5" width="191.25" height="33.75" align-x="150.75" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:matrix> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:Maximize auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve">profit</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:real>0.048625339429016104</ml:real> <ml:real>-0.0486252671320941</ml:real> </ml:matrix> </result> </ml:eval> </ml:define> </math> <rendering item-idref="59"/> </region> <region region-id="786" left="324" top="753" width="117.75" height="21" align-x="384" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">profit</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>28434.26450937325</ml:real> </result> </ml:eval> </math> <rendering item-idref="60"/> </region> <region region-id="812" left="324" top="777" width="79.5" height="15" align-x="370.5" align-y="786" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1.3850689103587377</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="15"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="61"/> </region> <region region-id="837" left="6" top="788.25" width="235.5" height="15" align-x="6" align-y="798" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Optimization subject to inequality constraint 0 <f family="Symbol" charset="2">  </f> s 2 . </text> </region> <region region-id="838" left="6" top="807" width="35.25" height="15" align-x="22.5" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="62"/> </region> <region region-id="839" left="54" top="807" width="35.25" height="15" align-x="70.5" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:real>0.3</ml:real> </ml:define> </math> <rendering item-idref="63"/> </region> <region region-id="840" left="108" top="807" width="24" height="12.75" align-x="119.25" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="64"/> </region> <region region-id="841" left="156" top="807" width="57.75" height="15" align-x="165.75" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> </ml:apply> </math> <rendering item-idref="65"/> </region> <region region-id="842" left="240" top="796.5" width="183" height="33.75" align-x="270.75" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:matrix> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:Maximize auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve">prod</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:real>0.27120827944648457</ml:real> <ml:real>0.057440613360656471</ml:real> </ml:matrix> </result> </ml:eval> </ml:define> </math> <rendering item-idref="66"/> </region> <region region-id="843" left="240" top="831" width="90" height="15" align-x="297" align-y="840" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">prod</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.0267902283728645</ml:real> </result> </ml:eval> </math> <rendering item-idref="67"/> </region> <region region-id="844" left="6" top="836.25" width="197.25" height="15" align-x="6" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Another way is to define "prod" only for 0<f family="Symbol" charset="2"></f>s2. </text> </region> <region region-id="800" left="348" top="837" width="69" height="15" align-x="394.5" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>-3.4161013643637352E-09</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="15"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="68"/> </region> <region region-id="845" left="6" top="852" width="177.75" height="47.25" align-x="63.75" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">prod</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve" subscript="2">p</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="69"/> </region> <region region-id="846" left="240" top="850.5" width="178.5" height="33.75" align-x="270.75" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:matrix> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:Maximize auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve">prod</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:real>0.27018788689117063</ml:real> <ml:real>0.060227279279471815</ml:real> </ml:matrix> </result> </ml:eval> </ml:define> <resultFormat numeric-only="true"> <general precision="4" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="70"/> </region> <region region-id="847" left="240" top="885" width="90" height="15" align-x="297" align-y="894" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> 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xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>2.475116187726743E-05</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="15"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="72"/> </region> <region region-id="857" left="6" top="900" width="217.5" height="47.25" align-x="66.75" align-y="918" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">profit</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="2">p</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:mult/> <ml:real>0.01</ml:real> <ml:id xml:space="preserve" subscript="f">s</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="73"/> </region> <region region-id="861" left="240" top="904.5" width="181.5" height="33.75" align-x="270.75" align-y="924" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:matrix> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:Maximize auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve">profit</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:real>0.2701874746171733</ml:real> <ml:real>0.060227267806441734</ml:real> </ml:matrix> </result> </ml:eval> </ml:define> <resultFormat numeric-only="true"> <general precision="4" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="74"/> </region> <region region-id="849" left="6" top="950.25" width="426" height="36" align-x="6" align-y="960" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Optimization finds a balance in allocating substrate for biomass production in the first fermentor and substrate for production production in the second fermentor. Substrate is all utilized at the end.<sp count="2"/>The optimum is right at the constraint boundary. </text> </region> <region region-id="896" left="0" top="990" width="6000" height="6" align-x="0" align-y="990" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="756" left="6" top="992.25" width="143.25" height="15" align-x="6" align-y="1002" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Effect of changing D1 at fixed D2. </text> </region> <region region-id="900" left="210" top="992.25" width="143.25" height="15" align-x="210" align-y="1002" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Effect of changing D2 at fixed D1. </text> 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push-down="false" lock-width="true"> Product formation parameters: </text> </region> <region region-id="403" left="12" top="105" width="34.5" height="15" align-x="27.75" align-y="114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="p">Y</ml:id> <ml:real>0.6</ml:real> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="409" left="54" top="98.25" width="19.5" height="27.75" align-x="62.25" align-y="114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:div/> <ml:id xml:space="preserve">g_P</ml:id> <ml:id xml:space="preserve">g_S</ml:id> </ml:apply> </math> <rendering item-idref="13"/> </region> <region region-id="405" left="90" top="105" width="38.25" height="15" align-x="105" align-y="114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="p">q</ml:id> <ml:real>0.02</ml:real> </ml:define> </math> <rendering item-idref="14"/> </region> <region region-id="411" left="138" top="98.25" width="39" height="27.75" align-x="150.75" align-y="114" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:div/> <ml:id xml:space="preserve">g_P</ml:id> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">g_cell</ml:id> <ml:id xml:space="preserve">hr</ml:id> </ml:apply> </ml:apply> </math> <rendering item-idref="15"/> </region> <region region-id="361" left="6" top="128.25" width="85.5" height="12" align-x="6" align-y="138" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Substrate feed rate: </text> </region> <region region-id="352" left="12" top="147" width="32.25" height="12.75" align-x="23.25" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">F</ml:id> <ml:real>100</ml:real> </ml:define> </math> <rendering item-idref="16"/> </region> <region region-id="421" left="48" top="140.25" width="18.75" height="27.75" align-x="55.5" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:div/> <ml:id xml:space="preserve">liter</ml:id> <ml:id xml:space="preserve">hr</ml:id> </ml:apply> </math> <rendering item-idref="17"/> </region> <region region-id="359" left="84" top="147" width="24.75" height="15" align-x="96.75" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="f">s</ml:id> <ml:real>5</ml:real> </ml:define> </math> <rendering item-idref="18"/> </region> <region region-id="422" left="114" top="140.25" width="18.75" height="27.75" align-x="121.5" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:div/> <ml:id xml:space="preserve">gm</ml:id> <ml:id xml:space="preserve">liter</ml:id> </ml:apply> </math> <rendering item-idref="19"/> </region> <region region-id="894" left="168" top="147" width="69" height="15" align-x="205.5" align-y="156" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="washout">D</ml:id> <ml:apply> <ml:id xml:space="preserve" subscript="1">μ</ml:id> <ml:id xml:space="preserve" subscript="f">s</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="20"/> </region> <region region-id="876" left="6" top="170.25" width="195.75" height="12" align-x="6" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Steady-state equations for the first fermentor. </text> </region> <region region-id="877" left="216" top="171" width="36.75" height="15" align-x="231.75" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">V</ml:id> <ml:real>500</ml:real> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="878" left="258" top="171" width="15.75" height="12.75" align-x="264.75" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">liter</ml:id> </math> <rendering item-idref="22"/> </region> <region region-id="879" left="294" top="164.25" width="65.25" height="30" align-x="310.5" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve">F</ml:id> <ml:id xml:space="preserve" subscript="1">V</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.2</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="887" left="360" top="165" width="21" height="18.75" align-x="369.75" align-y="180" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:pow/> <ml:id xml:space="preserve">hr</ml:id> <ml:real>-1</ml:real> </ml:apply> </math> <rendering item-idref="24"/> </region> <region region-id="681" left="6" top="189" width="25.5" height="15" align-x="19.5" align-y="198" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">x</ml:id> <ml:real>2</ml:real> </ml:define> </math> <rendering item-idref="25"/> </region> <region region-id="682" left="48" top="189" width="26.25" height="15" align-x="62.25" align-y="198" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">s</ml:id> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="26"/> </region> <region region-id="683" left="90" top="188.25" width="63.75" height="12" align-x="90" align-y="198" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> ... initial guess </text> </region> <region region-id="684" left="6" top="207" width="24" height="12.75" align-x="17.25" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="27"/> </region> <region region-id="663" left="18" top="218.25" width="26.25" height="15" align-x="18" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> dx1/dt </text> </region> <region region-id="659" left="60" top="219" width="84.75" height="15" align-x="69.75" align-y="228" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve" subscript="1">μ</ml:id> <ml:id xml:space="preserve" subscript="1">s</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="1">x</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="1">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="28"/> </region> <region region-id="665" left="18" top="242.25" width="26.25" height="15" align-x="18" align-y="252" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> ds1/dt </text> </region> <region region-id="666" left="60" top="236.25" width="126" height="30" align-x="69.75" align-y="252" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve" subscript="f">s</ml:id> <ml:id xml:space="preserve" subscript="1">s</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="s">Y</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="1">μ</ml:id> <ml:id xml:space="preserve" subscript="1">s</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="1">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="29"/> </region> <region region-id="426" left="6" top="273" width="90" height="15" align-x="42.75" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">xs1</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:Find auto-method="false" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve" subscript="1">x</ml:id> <ml:id xml:space="preserve" subscript="1">s</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="30"/> </region> <region region-id="688" left="126" top="273" width="74.25" height="15" 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subscript="1">s</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:indexer/> <ml:apply> <ml:id xml:space="preserve">xs1</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="32"/> </region> <region region-id="690" left="330" top="264.75" width="78.75" height="29.25" align-x="366" align-y="282" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">xs1</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:real>1.9199999999999995</ml:real> <ml:real>0.20000000000000015</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="33"/> </region> <region region-id="362" left="6" top="296.25" width="194.25" height="12" align-x="6" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Dynamic equations for the second fermentor. </text> </region> <region region-id="881" left="216" top="297" width="36.75" height="15" align-x="231.75" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">V</ml:id> <ml:real>300</ml:real> </ml:define> </math> <rendering item-idref="34"/> </region> <region region-id="882" left="258" top="297" width="15.75" height="12.75" align-x="264.75" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">liter</ml:id> </math> <rendering item-idref="35"/> </region> <region region-id="883" left="294" top="290.25" width="74.25" height="30" align-x="310.5" align-y="306" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:div/> <ml:id xml:space="preserve">F</ml:id> <ml:id xml:space="preserve" subscript="2">V</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.33333333333333331</ml:real> </result> </ml:eval> </ml:define> </math> <rendering item-idref="36"/> </region> <region region-id="892" left="372" top="297" width="21" height="18.75" align-x="381.75" align-y="312" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:pow/> <ml:id xml:space="preserve">hr</ml:id> <ml:real>-1</ml:real> </ml:apply> </math> <rendering item-idref="37"/> </region> <region region-id="691" left="6" top="315" width="25.5" height="15" align-x="19.5" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">x</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="38"/> </region> <region region-id="692" left="48" top="315" width="26.25" height="15" align-x="62.25" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="39"/> </region> <region region-id="693" left="96" top="315" width="27" height="15" align-x="111" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">p</ml:id> <ml:real>0</ml:real> </ml:define> </math> <rendering item-idref="40"/> </region> <region region-id="694" left="138" top="314.25" width="63.75" height="12" align-x="138" align-y="324" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> ... initial guess </text> </region> <region region-id="714" left="6" top="333" width="24" height="12.75" align-x="17.25" align-y="342" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="41"/> </region> <region region-id="706" left="18" top="344.25" width="26.25" height="15" align-x="18" align-y="354" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> dx2/dt </text> </region> <region region-id="695" left="66" top="345" width="129.75" height="15" align-x="75.75" align-y="354" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="1">x</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="2">x</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:id xml:space="preserve" subscript="2">μ</ml:id> <ml:id xml:space="preserve" subscript="2">s</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="2">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="42"/> </region> <region region-id="707" left="18" top="368.25" width="26.25" height="15" align-x="18" align-y="378" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> ds2/dt </text> </region> <region region-id="708" left="66" top="362.25" width="192" height="30" align-x="75.75" align-y="378" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:parens> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="1">s</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="2">s</ml:id> </ml:apply> </ml:parens> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="s">Y</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve" subscript="2">μ</ml:id> <ml:id xml:space="preserve" subscript="2">s</ml:id> </ml:apply> </ml:apply> <ml:id xml:space="preserve" subscript="2">x</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:mult/> <ml:apply> <ml:mult/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="p">Y</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="p">q</ml:id> </ml:apply> <ml:id xml:space="preserve" subscript="2">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="43"/> </region> <region region-id="713" left="18" top="392.25" width="26.25" height="15" align-x="18" align-y="402" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> ds2/dt </text> </region> <region region-id="711" left="66" top="393" width="69.75" height="15" align-x="75.75" align-y="402" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:equal/> <ml:real>0</ml:real> <ml:apply> <ml:minus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="p">q</ml:id> <ml:id xml:space="preserve" subscript="2">x</ml:id> </ml:apply> <ml:apply> <ml:mult/> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:id xml:space="preserve" subscript="2">p</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="44"/> </region> <region region-id="715" left="12" top="411" width="123.75" height="15" align-x="69" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">xsp2</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:Find auto-method="false" method="levenberg" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve" subscript="2">x</ml:id> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:id xml:space="preserve" subscript="2">p</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="45"/> </region> <region region-id="725" left="150" top="411" width="110.25" height="15" align-x="197.25" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:indexer/> <ml:apply> <ml:id xml:space="preserve">xsp2</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <ml:real>1</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="46"/> </region> <region region-id="726" left="282" top="411" width="111" height="15" align-x="330" align-y="420" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve" subscript="2">p</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:indexer/> <ml:apply> <ml:id xml:space="preserve">xsp2</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="47"/> </region> <region region-id="744" left="318" top="434.25" width="120" height="24" align-x="318" align-y="444" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> The value of s is 0.01X the value of p. </text> </region> <region region-id="718" left="12" top="432" width="110.25" height="47.25" align-x="69.75" align-y="450" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">prod</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve" subscript="2">p</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="48"/> </region> <region region-id="742" left="156" top="432" width="150" height="47.25" align-x="216.75" align-y="450" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">profit</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="2">p</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:mult/> <ml:real>0.01</ml:real> <ml:id xml:space="preserve" subscript="f">s</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="49"/> </region> <region region-id="740" left="156" top="495" width="90" height="15" align-x="213" align-y="504" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">prod</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.014399999999999998</ml:real> </result> </ml:eval> </math> <rendering item-idref="50"/> </region> <region region-id="741" left="258" top="481.5" width="120" height="51.75" align-x="314.25" align-y="510" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">xsp2</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="3" cols="1"> <ml:real>1.9199999999999995</ml:real> <ml:real>0.0080000000000001511</ml:real> <ml:real>0.11519999999999998</ml:real> </ml:matrix> </result> </ml:eval> </math> <rendering item-idref="51"/> </region> <region region-id="895" left="0" top="534" width="6000" height="6" align-x="0" align-y="534" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="745" left="6" top="537" width="72" height="15" align-x="22.5" align-y="546" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:range> <ml:sequence> <ml:real>0.01</ml:real> <ml:real>0.02</ml:real> </ml:sequence> <ml:real>1</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="52"/> </region> <region region-id="746" left="6" top="552" width="255" height="144" align-x="6" align-y="552" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="53"/> <rendering item-idref="54"/> </region> <region region-id="747" left="264" top="566.25" width="169.5" height="69" align-x="264" align-y="576" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Product concentration and product productivity continue to rise as D2 <f family="Symbol" charset="2"></f>0.<sp count="2"/>However, this is only because we failed to impose the constraint 0<f family="Symbol" charset="2"></f>s2.<sp count="2"/>Notice 0<f family="Symbol" charset="2"></f>s2 for small values of D2. </text> </region> <region region-id="777" left="6" top="698.25" width="456.75" height="15" align-x="6" align-y="708" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Optimization without constraint <f family="Symbol" charset="2">  </f> high product productivity, but unreal because of negative D 2 . </text> </region> <region region-id="780" left="12" top="723" width="35.25" height="15" align-x="28.5" align-y="732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="55"/> </region> <region region-id="781" left="60" top="723" width="35.25" height="15" align-x="76.5" align-y="732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:real>0.3</ml:real> </ml:define> </math> <rendering item-idref="56"/> </region> <region region-id="782" left="120" top="712.5" width="188.25" height="33.75" align-x="150.75" align-y="732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:matrix> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:Maximize auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve">prod</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:real>0.048625339429016104</ml:real> <ml:real>-0.0486252671320941</ml:real> </ml:matrix> </result> </ml:eval> </ml:define> </math> <rendering item-idref="57"/> </region> <region region-id="783" left="324" top="717" width="110.25" height="21" align-x="381" align-y="732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">prod</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>26799.049583556956</ml:real> </result> </ml:eval> </math> <rendering item-idref="58"/> </region> <region region-id="785" left="120" top="748.5" width="191.25" height="33.75" align-x="150.75" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:matrix> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:Maximize auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve">profit</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:real>0.048625339429016104</ml:real> <ml:real>-0.0486252671320941</ml:real> </ml:matrix> </result> </ml:eval> </ml:define> </math> <rendering item-idref="59"/> </region> <region region-id="786" left="324" top="753" width="117.75" height="21" align-x="384" align-y="768" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">profit</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>28434.26450937325</ml:real> </result> </ml:eval> </math> <rendering item-idref="60"/> </region> <region region-id="812" left="324" top="777" width="79.5" height="15" align-x="370.5" align-y="786" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>1.3850689103587377</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="15"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="61"/> </region> <region region-id="837" left="6" top="788.25" width="235.5" height="15" align-x="6" align-y="798" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Optimization subject to inequality constraint 0 <f family="Symbol" charset="2">  </f> s 2 . </text> </region> <region region-id="838" left="6" top="807" width="35.25" height="15" align-x="22.5" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:real>0.1</ml:real> </ml:define> </math> <rendering item-idref="62"/> </region> <region region-id="839" left="54" top="807" width="35.25" height="15" align-x="70.5" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:real>0.3</ml:real> </ml:define> </math> <rendering item-idref="63"/> </region> <region region-id="840" left="108" top="807" width="24" height="12.75" align-x="119.25" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">Given</ml:id> </math> <rendering item-idref="64"/> </region> <region region-id="841" left="156" top="807" width="57.75" height="15" align-x="165.75" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> </ml:apply> </math> <rendering item-idref="65"/> </region> <region region-id="842" left="240" top="796.5" width="183" height="33.75" align-x="270.75" align-y="816" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:matrix> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:Maximize auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve">prod</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:real>0.27120827944648457</ml:real> <ml:real>0.057440613360656471</ml:real> </ml:matrix> </result> </ml:eval> </ml:define> </math> <rendering item-idref="66"/> </region> <region region-id="843" left="240" top="831" width="90" height="15" align-x="297" align-y="840" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">prod</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.0267902283728645</ml:real> </result> </ml:eval> </math> <rendering item-idref="67"/> </region> <region region-id="844" left="6" top="836.25" width="197.25" height="15" align-x="6" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Another way is to define "prod" only for 0<f family="Symbol" charset="2"></f>s2. </text> </region> <region region-id="800" left="348" top="837" width="69" height="15" align-x="394.5" align-y="846" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>-3.4161013643637352E-09</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="15"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="68"/> </region> <region region-id="845" left="6" top="852" width="177.75" height="47.25" align-x="63.75" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">prod</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve" subscript="2">p</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="69"/> </region> <region region-id="846" left="240" top="850.5" width="178.5" height="33.75" align-x="270.75" align-y="870" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:matrix> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:Maximize auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve">prod</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:real>0.27018788689117063</ml:real> <ml:real>0.060227279279471815</ml:real> </ml:matrix> </result> </ml:eval> </ml:define> <resultFormat numeric-only="true"> <general precision="4" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="70"/> </region> <region region-id="847" left="240" top="885" width="90" height="15" align-x="297" align-y="894" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">prod</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.026780066441375667</ml:real> </result> </ml:eval> </math> <rendering item-idref="71"/> </region> <region region-id="850" left="348" top="885" width="63.75" height="15" align-x="394.5" align-y="894" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>2.475116187726743E-05</ml:real> </result> </ml:eval> <resultFormat numeric-only="true"> <general precision="3" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="15"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="72"/> </region> <region region-id="857" left="6" top="900" width="217.5" height="47.25" align-x="66.75" align-y="918" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:function> <ml:id xml:space="preserve">profit</ml:id> <ml:boundVars> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult/> <ml:parens> <ml:apply> <ml:lessOrEqual/> <ml:real>0</ml:real> <ml:apply> <ml:id xml:space="preserve" subscript="2">s</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> </ml:apply> </ml:parens> <ml:apply> <ml:div/> <ml:apply> <ml:minus/> <ml:apply> <ml:id xml:space="preserve" subscript="2">p</ml:id> <ml:sequence> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:mult/> <ml:real>0.01</ml:real> <ml:id xml:space="preserve" subscript="f">s</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="1">D</ml:id> </ml:apply> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:apply> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="73"/> </region> <region region-id="861" left="240" top="904.5" width="181.5" height="33.75" align-x="270.75" align-y="924" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:matrix> <ml:eval placeholderMultiplicationStyle="default"> <ml:apply> <ml:Maximize auto-method="true" method="conjugate" derivative-estimation="central" variable-estimation="tangent" linear-check="false" multistart="false" evolutionary="false"/> <ml:sequence> <ml:id xml:space="preserve">profit</ml:id> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve" subscript="2">D</ml:id> </ml:sequence> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:matrix rows="2" cols="1"> <ml:real>0.2701874746171733</ml:real> <ml:real>0.060227267806441734</ml:real> </ml:matrix> </result> </ml:eval> </ml:define> <resultFormat numeric-only="true"> <general precision="4" show-trailing-zeros="false" radix="dec" complex-threshold="10" zero-threshold="15" imaginary-value="i" exponential-threshold="3"/> <matrix display-style="auto" expand-nested-arrays="false"/> <unit format-units="false" simplify-units="false" fractional-unit-exponent="false"/> </resultFormat> </math> <rendering item-idref="74"/> </region> <region region-id="849" left="6" top="950.25" width="426" height="36" align-x="6" align-y="960" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Optimization finds a balance in allocating substrate for biomass production in the first fermentor and substrate for production production in the second fermentor. Substrate is all utilized at the end.<sp count="2"/>The optimum is right at the constraint boundary. </text> </region> <region region-id="896" left="0" top="990" width="6000" height="6" align-x="0" align-y="990" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <pageBreak/> </region> <region region-id="756" left="6" top="992.25" width="143.25" height="15" align-x="6" align-y="1002" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Effect of changing D1 at fixed D2. </text> </region> <region region-id="900" left="210" top="992.25" width="143.25" height="15" align-x="210" align-y="1002" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Effect of changing D2 at fixed D1. </text> </region> <region region-id="754" left="6" top="1011" width="99" height="15" align-x="22.5" align-y="1020" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:range> <ml:sequence> <ml:real>0.01</ml:real> <ml:real>0.02</ml:real> </ml:sequence> <ml:id xml:space="preserve" subscript="washout">D</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="75"/> </region> <region region-id="901" left="210" top="1011" width="72" height="15" align-x="226.5" align-y="1020" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:range> <ml:sequence> <ml:real>0.01</ml:real> <ml:real>0.02</ml:real> </ml:sequence> <ml:real>1</ml:real> </ml:range> </ml:define> </math> <rendering item-idref="76"/> </region> <region region-id="625" left="6" top="1026" width="195.75" height="186" align-x="6" align-y="1026" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="77"/> <rendering item-idref="78"/> </region> <region region-id="902" left="234" top="1026" width="200.25" height="186" align-x="234" align-y="1026" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="79"/> <rendering item-idref="80"/> </region> <region region-id="851" left="6" top="1226.25" width="33.75" height="12" align-x="6" align-y="1236" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> 3-D plot </text> </region> <region region-id="500" left="12" top="1245" width="30" height="12.75" align-x="25.5" align-y="1254" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">ni</ml:id> <ml:real>30</ml:real> </ml:define> </math> <rendering item-idref="81"/> </region> <region region-id="498" left="66" top="1245" width="35.25" height="12.75" align-x="74.25" align-y="1254" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">i</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">ni</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="82"/> </region> <region region-id="503" left="126" top="1244.25" width="46.5" height="27.75" align-x="150.75" align-y="1260" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="1">ΔD</ml:id> <ml:apply> <ml:div/> <ml:real>0.3</ml:real> <ml:id xml:space="preserve">ni</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="83"/> </region> <region region-id="507" left="192" top="1251" width="91.5" height="18.75" align-x="210.75" align-y="1260" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve" subscript="1">ΔD</ml:id> </ml:apply> <ml:real>0.00001</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="84"/> </region> <region region-id="502" left="12" top="1275" width="30" height="12.75" align-x="25.5" align-y="1284" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">nj</ml:id> <ml:real>30</ml:real> </ml:define> </math> <rendering item-idref="85"/> </region> <region region-id="499" left="66" top="1275" width="36" height="12.75" align-x="75" align-y="1284" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">j</ml:id> <ml:range> <ml:real>0</ml:real> <ml:id xml:space="preserve">nj</ml:id> </ml:range> </ml:define> </math> <rendering item-idref="86"/> </region> <region region-id="504" left="126" top="1274.25" width="46.5" height="27.75" align-x="150.75" align-y="1290" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve" subscript="2">ΔD</ml:id> <ml:apply> <ml:div/> <ml:real>0.5</ml:real> <ml:id xml:space="preserve">nj</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="87"/> </region> <region region-id="508" left="192" top="1281" width="93" height="18.75" align-x="211.5" align-y="1290" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> <ml:apply> <ml:plus/> <ml:apply> <ml:mult/> <ml:id xml:space="preserve">j</ml:id> <ml:id xml:space="preserve" subscript="2">ΔD</ml:id> </ml:apply> <ml:real>0.00001</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="88"/> </region> <region region-id="899" left="30" top="1305" width="105" height="18.75" align-x="70.5" align-y="1314" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">PROD</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">prod</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="89"/> </region> <region region-id="898" left="240" top="1305" width="115.5" height="18.75" align-x="288" align-y="1314" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve">PROFIT</ml:id> <ml:sequence> <ml:id xml:space="preserve">i</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">profit</ml:id> <ml:sequence> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="1">D</ml:id> <ml:id xml:space="preserve">i</ml:id> </ml:apply> <ml:apply> <ml:indexer/> <ml:id xml:space="preserve" subscript="2">D</ml:id> <ml:id xml:space="preserve">j</ml:id> </ml:apply> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="90"/> </region> <region region-id="903" left="6" top="1332" width="181.5" height="204" align-x="96" align-y="1332" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="013500E0-1122-11DB-9380-000D56C6051A" item-idref="91" disable-calc="false"> <inputs> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">PROD</ml:id> </inputs> <outputs/> </component> <rendering item-idref="92"/> </region> <region region-id="523" left="222" top="1332" width="181.5" height="204" align-x="312" align-y="1332" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <component hide-arguments="false" clsid-buddy="013500E0-1122-11DB-9380-000D56C6051A" item-idref="93" disable-calc="false"> <inputs> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math30">PROFIT</ml:id> </inputs> <outputs/> </component> <rendering item-idref="94"/> </region> <region region-id="904" left="6" top="1545" width="85.5" height="12.75" align-x="58.5" align-y="1554" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval placeholderMultiplicationStyle="default" xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:apply> <ml:id xml:space="preserve">max</ml:id> <ml:id xml:space="preserve">PROD</ml:id> </ml:apply> <result xmlns="http://schemas.mathsoft.com/math30"> <ml:real>0.026302222837983127</ml:real> </result> </ml:eval> </math> <rendering item-idref="95"/> </region> <region region-id="905" left="6" top="1563" width="147" height="18" align-x="33" align-y="1572" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math30"> <ml:id xml:space="preserve">index</ml:id> <ml:apply> <ml:indexer/> <ml:apply> <ml:id xml:space="preserve">match</ml:id> <ml:sequence> <ml:apply> <ml:id xml:space="preserve">max</ml:id> <ml:id xml:space="preserve">PROD</ml:id> </ml:apply> <ml:id xml:space="preserve">PROD</ml:id> </ml:sequence> </ml:apply> <ml:real>0</ml:real> </ml:apply> </ml:define> </math> <rendering item-idref="96"/> </region> <region region-id="906" left="132" top="1592.25" width="38.25" height="15" align-x="132" align-y="1602" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> D1 & D2 <sp/> </text> </region> <region region-id="907" left="6" top="1582.5" width="117" height="39.75" 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