cbom.atozmath.com
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104.21.61.249
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Submitted URL: http://cbom.atozmath.com/
Effective URL: http://cbom.atozmath.com/Menu/CBomMenu.aspx
Submission: On October 17 via api from US — Scanned from DE
Effective URL: http://cbom.atozmath.com/Menu/CBomMenu.aspx
Submission: On October 17 via api from US — Scanned from DE
Form analysis 2 forms found in the DOM
POST ./CBomMenu.aspx
<form method="post" action="./CBomMenu.aspx" id="form1">
<div class="aspNetHidden">
<input type="hidden" name="__VIEWSTATE" id="__VIEWSTATE"
value="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">
</div>
<div class="aspNetHidden">
<input type="hidden" name="__VIEWSTATEGENERATOR" id="__VIEWSTATEGENERATOR" value="63907D12">
<input type="hidden" name="__EVENTTARGET" id="__EVENTTARGET" value="">
<input type="hidden" name="__EVENTARGUMENT" id="__EVENTARGUMENT" value="">
<input type="hidden" name="__EVENTVALIDATION" id="__EVENTVALIDATION" value="/wEdAAVnVhDbGcQvw1zRx7QP97DbOWTnjwhmBXYiezH5kDQFYnj/rwfbfGkaRxKMK9xtlWq04xyZ5alu8QPxhoNSjZm6PVLjCljcvgQY24QSdIotnrYCJd5ImtigV8T0zn6FCJCRD+Uu">
</div>
<table cellpadding="0" cellspacing="0" width="100%">
<tbody>
<tr class="menubg">
<td>
<div class="adsbyvli" data-ad-slot="vi_az" data-width="160" style="width: 160px; height: 77px; display: none;"></div>
</td>
</tr>
</tbody>
</table>
<table id="Table1" cellpadding="0" cellspacing="0" class="table0" width="100%" style="width:100%;min-width: 305px;">
<tbody>
<tr class="menubg" align="center">
<td>
<table id="Table4" cellpadding="0" cellspacing="0" class="table0" width="100%">
<tbody>
<tr class="menubg">
<td valign="middle">
<div id="consentframe" style="width: 100%; color: black; display: table;">
<div style="border: 1px solid gray; background: #eeeefe; padding: 0px; margin: 0px 0px 0px 0px; text-align: justify; font-size: 14px;">
<p style="vertical-align: middle;"> We use cookies to improve your experience on our site and to show you relevant advertising. By browsing this website, you agree to our use of cookies. <a href="//atozmath.com/privacy.aspx">Learn
more</a> <input type="button" id="cookiesok" title="Agree" value="Accept">
</p>
</div>
</div>
<script>
var allowcookies = null;
var v = new RegExp("cookiesconsent=([^;]+)").exec(document.cookie);
if (v != null) {
allowcookies = unescape(v[1]);
};
var bar = document.getElementById('consentframe'),
btn = document.getElementById('cookiesok');
btn.onclick = function() {
bar.style.display = 'none';
var d = new Date;
d.setTime(d.getTime() + 24 * 60 * 60 * 1000 * 365);
document.cookie = 'cookiesconsent=ok;path=/;expires=' + d.toGMTString();
};
bar.style.display = (allowcookies == 'ok') ? 'none' : 'table';
</script>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr class="menubg">
<td>
<script language="javascript" type="text/javascript">
function HC_KeyBoard_Click() {
var oWindow = 0;
if (oWindow) {
if (!oWindow.closed) oWindow.close();
};
oWindow = window.open('//atozmath.com/Keyboard.aspx?show=1', 'Keyboard', 'status=no,width=520,height=420,resizable=1,scrollbars=1');
oWindow.focus();
return false;
}
</script>
<table id="Table1" cellpadding="0" cellspacing="0" width="100%">
<tbody>
<tr class="menubg1">
<td>
<table id="Table2" cellpadding="0" cellspacing="0" width="100%">
<tbody>
<tr class="menubg1">
<td width="25%" valign="middle" align="left">
<a title="AtoZmath.com" href="//atozmath.com">
<img src="//cbom.atozmath.com/Style/logonew.jpg" alt="AtoZmath.com" width="100%">
</a>
</td>
<td valign="top">
<table id="Table3" cellpadding="0" cellspacing="0" width="100%" class="table0">
<tbody>
<tr>
<td>
<table width="100%" border="0">
<tbody>
<tr>
<td align="center" width="50%">
</td>
<td align="center" valign="middle">
<a class="footer1" href="//atozmath.com/Help/SupportUs.aspx" title="Support us">
Support us </a>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr style="display:none">
<td align="center">
<font size="3">
<b><font color="red">(New)</font> All problem can be solved using search box</b>
</font>
</td>
</tr>
<tr>
<td align="center">
<a href="//atozmath.com/Help/Aboutus.aspx">
<font size="2">
<strike>I want to sell my website www.AtoZmath.com with complete code</strike>
</font>
</a>
</td>
</tr>
<tr>
<td align="center">
<a href="Javascript:void(0)" onclick="HC_KeyBoard_Click();" title="Input Help">
<img src="//atozmath.com/IMAGES1/KeyBoard.png">
</a>
</td>
</tr>
</tbody>
</table>
</td>
<td width="25%" valign="middle" align="center">
<table id="AutoNumber10" style="border-collapse: collapse" cellspacing="0" cellpadding="0" width="100%" border="0">
<tbody>
<tr>
<td align="center">
<div id="google_translate_element">
<div class="skiptranslate goog-te-gadget" dir="ltr" style="display: none;">
<div id=":0.targetLanguage"><select class="goog-te-combo" aria-label="Widget "Sprache übersetzen""></select></div>Powered by <span
style="white-space:nowrap"><a class="VIpgJd-ZVi9od-l4eHX-hSRGPd" href="https://translate.google.com" target="_blank"><img src="https://www.gstatic.com/images/branding/googlelogo/1x/googlelogo_color_42x16dp.png" width="37px" height="14px" style="padding-right: 3px" alt="Google Google Übersetzer">Google Übersetzer</a></span>
</div>
</div>
<script>
function googleTranslateElementInit() {
new google.translate.TranslateElement({
pageLanguage: 'en'
}, 'google_translate_element');
}
</script>
<script src="//translate.google.com/translate_a/element.js?cb=googleTranslateElementInit"></script>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr class="menubg1">
<td align="center" width="100%">
<script language="javascript" type="text/javascript" src="//atozmath.com/Scripts/actb/actb_AZ.js"></script>
<script language="javascript" type="text/javascript" src="//atozmath.com/Scripts/actb/common.js"></script>
<div id="HeaderCtl1_SerarchSolveCtl1_pnlSearch" onkeypress="javascript:return WebForm_FireDefaultButton(event, 'HeaderCtl1_SerarchSolveCtl1_BtnFind')">
<table id="HeaderCtl1_SerarchSolveCtl1_tblSearch" style="border-collapse: collapse" cellspacing="0" cellpadding="1" border="0" width="90%">
<tbody>
<tr>
<td align="right" width="99%">
<input name="ctl00$HeaderCtl1$SerarchSolveCtl1$txt1" type="text" id="HeaderCtl1_SerarchSolveCtl1_txt1" placeholder="Try our new - Enter problem or search problem" autocomplete="off" onkeyup="return txt1_Keypress2();"
style="font-size:Medium;width:99%;">
</td>
<td align="left">
<input type="image" name="ctl00$HeaderCtl1$SerarchSolveCtl1$BtnFind" id="HeaderCtl1_SerarchSolveCtl1_BtnFind" src="../IMAGES1/Search2.png" onclick="return BtnFind2_Click();" style="height:25px;">
<input name="ctl00$HeaderCtl1$SerarchSolveCtl1$hQId" type="hidden" id="HeaderCtl1_SerarchSolveCtl1_hQId">
<div id="HeaderCtl1_SerarchSolveCtl1_divExtraQueOptions" style="display:none"></div>
</td>
</tr>
</tbody>
</table>
</div>
<script language="javascript" type="text/javascript" id="JsLoad">
var obj2;
function textFilter2() {
var customarray = [];
obj2 = new actb(document.getElementById('HeaderCtl1_SerarchSolveCtl1_txt1'), customarray);
obj2.actb_delimiter = new Array();
obj2.actb_startcheck = 2;
obj2.hQId = document.getElementById('HeaderCtl1_SerarchSolveCtl1_hQId');
obj2.divExtraQueOptions = document.getElementById('HeaderCtl1_SerarchSolveCtl1_divExtraQueOptions');
obj2.preview = 0;
}
function txt1_Keypress2() {
var x = document.getElementById('HeaderCtl1_SerarchSolveCtl1_txt1').value;
if (x != "" && x.length == 2) {
$.ajax({
type: "POST",
url: "//cbom.atozmath.com/WebMethods.aspx/getProblems",
data: '{S2: "' + x + '" }',
contentType: "application/json; charset=utf-8",
dataType: "json",
success: function(result) {
//alert(x +" => "+ result);
var customarray = JSON.stringify(result).split("~");
obj2.actb_keywords = customarray;
document.getElementById('HeaderCtl1_SerarchSolveCtl1_BtnFind').focus();
document.getElementById('HeaderCtl1_SerarchSolveCtl1_txt1').focus();
},
failure: function(result) {
alert("failure : " + final_transcript + "=>" + result);
}
});
}
}
</script>
<script language="javascript" type="text/javascript">
<!--
function BtnFind2_Click() {
if (isBlankCtlStr(document.getElementById("HeaderCtl1_SerarchSolveCtl1_txt1"))) return false;
//relace 1<x<7 to 1< x< 7 because <x gives error
//document.getElementById("HeaderCtl1_SerarchSolveCtl1_txt1").value = document.getElementById("HeaderCtl1_SerarchSolveCtl1_txt1").value.replace("<","< ");
document.getElementById("HeaderCtl1_SerarchSolveCtl1_txt1").value = document.getElementById("HeaderCtl1_SerarchSolveCtl1_txt1").value.replace(/<([a-zA-Z])/, "< $1");
var Sq1 = getContentS2();
//
-->
Common
code
var
SQS
=
"q1=" +
encodeURIComponent(Sq1);
SQS
+=
"&do=1";
//
<--Common code location.href="//atozmath.com/Default.aspx?" + SQS; return false; } function getContentS2() { var SVal=document.getElementById("HeaderCtl1_SerarchSolveCtl1_txt1").value; SVal +="`" +
document.getElementById("HeaderCtl1_SerarchSolveCtl1_hQId").value; return SVal; } //to set list textFilter2(); -->
</script>
</td>
</tr>
<tr class="menubg1">
<td align="right">
<table id="Table2" border="0" cellpadding="0" cellspacing="0" width="90%">
<tbody>
<tr>
<td>
<a class="topmenu5" href="//atozmath.com/default.aspx" title="Home">
Home</a>
</td>
<td>
<a class="topmenu5" href="//atozmath.com/Menu/WhatsNewMenu.aspx" title="What's new">
What's new</a>
</td>
<td>
<a class="topmenu5" href="//atozmath.com/Menu/CollegeAlgebra.aspx" title="College Algebra">
College Algebra</a>
</td>
<td>
<a class="topmenu5" href="//atozmath.com/Menu/GamesMenu.aspx" title="All Available Games">
Games</a>
</td>
<td>
<a class="topmenu5" href="//atozmath.com/Feedback/Feedback.aspx" title="Feedback">Feedback</a>
</td>
<td>
<a class="topmenu5" href="//atozmath.com/Help/Aboutus.aspx" title="About us">
About us</a>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr bgcolor="#e2f1ff">
<td align="left">
<div align="left" class="menuBarBorder">
<table id="Table1" cellpadding="0" cellspacing="0" width="100%">
<tbody>
<tr bgcolor="#e2f1ff">
<td align="left" colspan="2">
<div align="left" class="menuBarBorder">
<table id="Table3" border="0" cellpadding="0" cellspacing="0" width="100%">
<tbody>
<tr>
<td>
<a class="topmenu5" href="//atozmath.com/Menu/Algebra.aspx" title="Algebra">
Algebra</a>
</td>
<td>
<a class="topmenu5" href="//atozmath.com/Menu/MatrixAlgebra.aspx" title="Matrix & Vector">
Matrix & Vector</a>
</td>
<td>
<a class="topmenu5" href="//atozmath.com/Menu/ConmMenu.aspx" title="Numerical Methods">
Numerical Methods</a>
</td>
<td>
<a class="topmenu5" href="//atozmath.com/Menu/StatisticsMenu.aspx" title="Statistical Methods">
Statistical Methods</a>
</td>
<td>
<a class="topmenu5" href="//cbom.atozmath.com/Menu/CBomMenu.aspx" title="Operation Research">
Operation Research</a>
</td>
<td>
<a class="topmenu5" href="//atozmath.com/Menu/WordingProblems.aspx" title="Word Problems">
Word Problems</a>
</td>
<td>
<a class="topmenu5" href="//atozmath.com/Menu/CalculusMenu.aspx" title="Calculus">
Calculus</a>
</td>
<td>
<a class="topmenu5" href="//atozmath.com/Menu/PlotsGeometry.aspx" title="Geometry">
Geometry</a>
</td>
<td>
<a class="topmenu5" href="//atozmath.com/Menu/PreAlgebra.aspx" title="Pre-Algebra">
Pre-Algebra</a>
</td>
</tr>
</tbody>
</table>
</div>
</td>
</tr>
</tbody>
</table>
</div>
</td>
</tr>
</tbody>
</table>
<script language="javascript" type="text/javascript">
var ServerURL = "//atozmath.com";
</script>
</td>
</tr>
<tr valign="top">
<td>
<table id="FkuiNTop" cellpadding="0" cellspacing="0" class="table0" width="100%">
<tbody>
<tr valign="top">
<td class="menubg" valign="top" style="width:301px;">
<table id="tFkuiN1_1" border="0" cellpadding="1" cellspacing="0" width="301px">
<tbody>
<tr>
<td>
<div id="FkuiN1_1"></div>
<div id="FkuiN1_2" style="background-color:;width:300px;height:230px;">
<div style="width:296px">
<div class="adsbyvli" data-ad-slot="vi_31308697" data-width="300" data-height="250" id="vi_31308697"
style="width: 300px; margin: 0px auto !important; position: relative !important; visibility: visible !important; opacity: 1 !important; display: block !important; height: 230px; max-height: 230px !important;">
<vli id="ifr_vi_31308697"
style="left: -420px; transition: left 0.8s ease 0s; display: block !important; overflow: hidden !important; transform: scale(1) !important; z-index: 2147483647 !important; width: 300px !important; height: 230px !important;">
<vli class="playerWrapper videoDiscovery" style="position: absolute; top: 0px; left: 0px; height: 100%; width: 100%;">
<div id="vi__disable__fixed" style="margin: 5px 5px 8px; display: none;">
<vli id="vi__disable_fixed_btn_icon"
style="line-height: 1;cursor: pointer;position: relative;width: 16px; height: 16px; background-color: rgb(255 255 255 / 50%); box-shadow: rgb(0, 0, 0) 0px 0px 3px 0px; border-radius: 16px; display: block;">
<div style="pointer-events: auto; width: 60px;height: 60px;position: absolute;top: 50%;left: 50%;transform: translate(-50%, -50%);z-index: -1;"></div><svg style="margin-bottom: 8px" viewBox="0 0 128 128"
xmlns="http://www.w3.org/2000/svg">
<title></title>
<path d="M64,0a64,64,0,1,0,64,64A64.07,64.07,0,0,0,64,0Zm0,122a58,58,0,1,1,58-58A58.07,58.07,0,0,1,64,122Z"></path>
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</svg>
</vli>
</div>
<style>
.vi__slider #vi__top {
position: relative;
height: auto !important
}
.vi__slider #vi__content {
width: calc(100% + 2px);
height: 99px;
position: relative;
padding-top: 7px
}
.vi__slider .vi__flx {
height: 100%
}
.vi__slider .vi__post {
margin-right: 7px;
height: 100%
}
.vi__slider .vi__post a {
position: relative;
display: block;
height: 100%
}
.vi__slider .vi__thumbctn {
height: 100%
}
.vi__slider .vi__title {
position: absolute;
top: 1px;
left: 3px;
width: calc(100% - 6px);
text-align: center;
display: none;
height: auto !important
}
.vi__slider #vi__content .vi__flx:hover .vi__title {
display: block
}
.vi__slider #vi__content .vi__flx:hover .vi__mask {
background-color: rgba(0, 0, 0, .4);
-webkit-transition: background-color .6s linear;
-ms-transition: background-color .6s linear;
transition: background-color .6s linear
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.vi__slider .vi__title span {
color: #fff;
display: block;
font-size: 13px;
overflow: hidden;
text-overflow: ellipsis;
white-space: nowrap;
width: 100%
}
.vi__slider .vi__featured .vi__title span {
display: none
}
.vi__slider .vi__mask {
width: 100%;
height: 100%;
position: absolute;
top: 0;
bottom: 0;
left: 0;
right: 0;
background: 0 0
}
.vi__slider .vi__mask:hover>svg {
display: block
}
.vi__slider .vi__featured .vi__mask>svg {
display: none
}
.vi__slider .vi__featured .vi__mask {
background-color: rgba(0, 0, 0, .4)
}
.vi__slider .vi__mask>svg {
display: none;
width: 30px;
height: 30px;
position: absolute;
top: 50%;
left: 50%;
transform: translate(-50%, -28%)
}
.vi__slider .vi__featured a:after {
content: 'Now\aPlaying';
position: absolute;
white-space: pre-wrap;
color: #fff;
text-transform: uppercase;
top: 50%;
left: 50%;
transform: translate(-50%, -50%);
text-align: center;
font-size: 12px
}
.vi__slider .siema_next,
.vi__slider .siema_prev {
width: 10px;
height: 16px;
top: 50%;
position: absolute;
transform: translateY(-50%);
cursor: pointer;
padding: 5px;
box-sizing: initial !important
}
.vi__slider .siema_next:hover,
.vi__slider .siema_prev:hover {
background: rgba(0, 0, 0, .5)
}
.vi__slider .siema_prev {
left: 7px
}
.vi__slider .siema_next {
right: 10px
}
.player-fixed #vi__content {
display: none
}
.playerWrapper.videoDiscovery {
margin: 0;
padding: 0;
font-family: Source Sans Pro, sans-serif
}
#vi__wrapper img {
width: 100%;
height: auto
}
#vi__adcontainer {
width: 100%;
height: 506.25px;
position: relative;
overflow: hidden
}
#vi__adplayer {
width: 100%;
height: 100%;
position: relative
}
.vi__thumb {
width: 100%;
height: 100%;
background-position: center;
background-size: cover;
background-repeat: no-repeat;
position: relative
}
#vi__toggle_fullscreen {
display: none;
flex-direction: row-reverse;
height: auto
}
#vi__toggle_fullscreen_btn {
width: 105px;
border-bottom: 20px solid rgba(0, 0, 0, .7);
border-left: 10px solid transparent;
position: relative
}
#vi__toggle_fullscreen_btn vli {
position: absolute;
top: 0;
left: 0;
height: 16px;
display: flex;
padding: 2px
}
#vi__toggle_fullscreen_icon {
cursor: pointer
}
#vi__btn_pbs {
line-height: 1;
border-top: 1px solid #e6e6e663;
background-color: #fff;
position: absolute;
left: 50%;
transform: translate(-50%, 0);
bottom: -26px;
height: 25px;
width: 100%;
display: flex;
justify-content: center;
z-index: 2
}
.vi__player {
width: 100%;
height: 100%;
background-color: #eee
}
video {
width: 100%;
height: 100% !important
}
[class*=" vi-"],
[class^=vi-] {
font-family: "Helvetica Neue", "Calibri Light", Roboto, sans-serif !important;
speak: none;
font-style: normal;
font-weight: 400;
font-variant: normal;
text-transform: none;
line-height: 1;
-webkit-font-smoothing: antialiased;
-moz-osx-font-smoothing: grayscale
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border: 0 solid rgba(0, 0, 0, .2);
box-shadow: 0 0 20px rgba(0, 0, 0, .2);
position: relative;
overflow: hidden;
font-family: "Helvetica Neue", "Calibri Light", Roboto, sans-serif;
letter-spacing: .02em
}
.vi__player.vi__fullscreen {
position: fixed;
top: 0;
left: 0;
right: 0;
bottom: 0;
height: 100%;
width: 100%;
z-index: 10000000;
background: #000;
border-radius: 0 !important;
display: -ms-flexbox;
display: flex
}
.lds-dual-ring {
display: none;
width: 40px;
height: 40px !important;
position: absolute;
top: 50%;
left: 50%;
transform: translate(-50%, -50%)
}
.lds-dual-ring:after {
content: " ";
display: none;
width: 25px;
height: 25px;
margin: 1px;
border-radius: 50%;
border: 5px solid #fff;
border-color: #fff transparent #fff transparent;
animation: lds-dual-ring 1.2s linear infinite;
-webkit-animation: lds-dual-ring 1.2s linear infinite;
-moz-animation: lds-dual-ring 1.2s linear infinite;
-o-animation: lds-dual-ring 1.2s linear infinite
}
.vi__loading .lds-dual-ring,
.vi__loading .lds-dual-ring:after {
display: inline-block
}
@keyframes lds-dual-ring {
0% {
transform: rotate(0);
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}
100% {
transform: rotate(360deg);
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@-webkit-keyframes lds-dual-ring {
0% {
transform: rotate(0);
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100% {
transform: rotate(360deg);
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}
@-moz-keyframes lds-dual-ring {
0% {
transform: rotate(0);
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}
100% {
transform: rotate(360deg);
-ms-transform: rotate(360deg)
}
}
@-o-keyframes lds-dual-ring {
0% {
transform: rotate(0);
-ms-transform: rotate(0)
}
100% {
transform: rotate(360deg);
-ms-transform: rotate(360deg)
}
}
.default__control button {
background: no-repeat;
border: none;
outline: 0;
padding: 0;
display: inline-block;
cursor: pointer
}
.default__control.vi_control_overlay {
position: absolute;
top: 0;
bottom: 0;
left: 0;
right: 0;
background-color: rgba(0, 0, 0, .5)
}
.default__control {
opacity: 0;
-webkit-transition: opacity .3s ease-in-out;
-moz-transition: opacity .3s ease-in-out;
-ms-transition: opacity .3s ease-in-out;
-o-transition: opacity .3s ease-in-out;
transition: opacity .3s ease-in-out
}
.vi__player:hover .default__control {
opacity: 1
}
.default__control.vi__control__top {
position: absolute;
top: 0;
left: 0;
width: 100%
}
.vi__control__top #vi__video_title {
text-align: left !important
}
.vi__button {
font-size: 14px !important
}
.vi__control__top #vi__video_title a {
color: #fff;
text-decoration: none;
font-size: 15px;
margin: 10px;
display: inline-block;
white-space: nowrap;
overflow: hidden;
width: calc(100% - 150px);
text-overflow: ellipsis
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.default__control.vi__control__middle {
position: absolute;
top: 50%;
left: 50%;
transform: translate(-50%, -50%)
}
.default__control.vi__control__middle .vi__middle_button {
display: -ms-flexbox;
display: flex;
justify-content: center;
-ms-flex-align: center;
align-items: center;
justify-content: center;
align-items: center;
position: absolute;
top: 50%;
left: 50%;
-ms-transform: translate(-50%, -50%) scale(1);
transform: translate(-50%, -50%) scale(1)
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.default__control.vi__control__middle .vi__middle_button button {
width: 14px;
height: 14px;
margin: 0 5px
}
.default__control.vi__control__middle .vi__middle_button button.big {
width: 26px;
height: 26px
}
.vi__loading .vi__control__middle .vi__middle_button {
display: none
}
.default__control.vi__control__bottom {
position: absolute;
bottom: 0;
left: 0;
width: 100%;
height: 42px !important
}
.vi__control__bottom .progress {
margin-bottom: 8px;
position: relative;
height: 5px;
transition: height .3s;
background: rgba(0, 0, 0, .5);
cursor: pointer;
flex: none;
-ms-flex: none;
display: flex;
width: 100%
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.vi__control__bottom .progress .progress__filled {
width: 0%;
background: #ddd;
-ms-flex: 0;
flex: 0;
-ms-flex-preferred-size: 0;
flex-basis: 0%
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.vi__control__bottom button {
float: right;
margin-right: 10px;
position: static;
line-height: 6px;
width: auto;
height: auto
}
.vi__control__bottom .vi__time {
color: #ddd;
margin-left: 5px;
font-size: 13px;
display: inline-block
}
.vi__control__bottom .vi__time.vi__current_time {
margin-left: 10px
}
.vi__custom_logo {
position: absolute;
top: 10px;
right: 10px;
height: 20px;
max-width: 20px;
overflow: hidden;
-webkit-transition: max-width .5s ease-in-out;
transition: max-width .5s ease-in-out;
display: block
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.vi__custom_logo a {
display: -webkit-box;
height: 20px;
text-decoration: none;
color: #fff
}
.vi__custom_logo .vi__svg_content {
display: block;
width: 20px;
height: 20px;
overflow: hidden
}
.vi__custom_logo .vi__text_content {
margin-left: 5px;
font-size: 13px;
white-space: nowrap
}
.vi__player:hover .vi__custom_logo {
max-width: 150px
}
.vi__fullscreen .vi__custom_logo,
.vi__player_fixed .vi__custom_logo {
display: none
}
#vi_toggleVolVideoDiscovery {
padding: initial;
background-color: transparent;
z-index: 9999;
left: 10px;
bottom: 6px;
cursor: pointer
}
#vi_toggleVolVideoDiscovery svg {
font-size: initial;
padding: initial;
background-color: transparent;
max-width: 100%;
max-height: 100%;
width: 100%;
height: 100%
}
.sound_on .vi__mutedVideo {
display: none !important
}
.sound_off .vi__volVideo {
display: none !important
}
.videoDiscovery vli {
display: block;
height: 100%
}
.videoWrapper .videoDiscovery vli {
display: block;
height: initial
}
.vi__wgcontainer {
height: auto !important
}
</style>
<style>
.vi__slider .vi__featured a:after {
content: "Now Playing";
}
</style>
<vli id="vi__toggle_fullscreen">
<vli id="vi__toggle_fullscreen_btn" style="width: 35px; border-bottom: 20px solid rgba(0, 0, 0, 0.7); border-left: 10px solid transparent; position: relative;">
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<g fill="#fff">
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d="m373.3 228.67h-108.27c-0.036 0-0.06 0.012-0.084 0.012-0.036 0-0.06-0.012-0.096-0.012-6.713 0-12.03 5.317-12.03 12.03v108.27c0 6.833 5.39 11.922 12.223 11.934 6.821 0.012 11.838-5.101 11.838-11.922v-96.242h96.421c6.833 0 12.03-5.197 12.03-12.03s-5.196-12.031-12.03-12.043z"
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</g>
</svg> </span> </span> </vli>
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<vli id="vi__wrapper" class="vi__slider">
<vli class="vi__wgcontainer">
<vli id="vi__top" style="background-color: unset;">
<vli id="vi__adcontainer" style="height: 169px; width: 100%; margin: 0px auto;">
<vli id="vi__adplayer" style="z-index:9997">
<vli id="vi__player" style="width:100%;height:100%">
<vli class="vi__player vi__pc default is_playing vi__loading" style="position:relative"> <video preload="auto" playsinline="" muted=""
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<source src="//media.bilsyndication.com/vid/?id=abcd&t=y" type="video/mp4">
</video> <img id="vi_poster__videoDiscovery" class="vliIgnore" style="position: absolute; top: 0px; left: 0px; visibility: hidden; opacity: 0; width: inherit !important; height: inherit !important;"
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<vli class="default__control vi_control_overlay"></vli>
<vli class="default__control vi__control__top">
<vli id="vi__video_title" title="AtoZmath1">
<a rel="nofollow" href="" target="_blank" title="7 Easy Crafts Ideas at Home"> <span>AtoZmath1</span> <span><svg style="width:14px;height:14px;transform:scaleX(-1)" xmlns="http://www.w3.org/2000/svg" viewBox="-22 0 512 512.00667"> <path fill="#fff" d="M422.188 98.813h-44.91a8.986 8.986 0 0 0-8.985 8.984c0 4.957 4.023 8.98 8.984 8.98h44.91c14.856 0 26.95 12.09 26.95 26.95v323.367c0 14.86-12.094 26.949-26.95 26.949H98.82c-14.86 0-26.949-12.09-26.949-26.95V251.517c0-4.957-4.023-8.985-8.98-8.985-4.961 0-8.985 4.028-8.985 8.985v215.578c0 24.765 20.149 44.914 44.914 44.914h323.368c24.765 0 44.914-20.149 44.914-44.914V143.727c0-24.766-20.149-44.914-44.914-44.914zm0 0"> </path> <path fill="#fff" d="M156.566 250.645a8.971 8.971 0 0 0 5.121-8.114v-53.894h8.985c108.531 0 156.437 53.07 198.539 138.699a8.974 8.974 0 0 0 10.094 4.789 8.98 8.98 0 0 0 6.945-8.75c0-228.938-103.613-260.492-215.578-260.492h-8.984V8.988a8.979 8.979 0 0 0-5.122-8.11 8.939 8.939 0 0 0-9.52 1.142L3.329 118.79c-2.101 1.698-3.324 4.269-3.324 6.972s1.223 5.261 3.324 6.968l143.719 116.774a8.942 8.942 0 0 0 9.52 1.14zm0 0"> </path> </svg> </span> </a>
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<vli class="lds-dual-ring"></vli>
<vli class="default__control vi__control__middle">
<vli class="vi__middle_button"><button type="button" class="vi__button vi__button_prev" id="vi__prepost" title="Previous"><svg enable-background="new 0 0 42 42" version="1.1" viewBox="0 0 42 42"
xml:space="preserve" xmlns="http://www.w3.org/2000/svg">
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fill="#ddd"></path>
</svg></button> <button type="button" class="vi_button_play_pause toggle big" title="Play"><svg enable-background="new 0 0 232.153 232.153" version="1.1" viewBox="0 0 232.15 232.15"
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<path class="play_pause"
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clip-rule="evenodd" fill="#ddd" fill-rule="evenodd"></path>
</svg></button> <button type="button" class="vi__button vi__button_next" id="vi__nextpost" title="Next"><svg enable-background="new 0 0 42 42" version="1.1" viewBox="0 0 42 42"
xml:space="preserve" xmlns="http://www.w3.org/2000/svg">
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d="M35.5,0c-0.552,0-1,0.447-1,1v18.095L7.068,0.177C6.762-0.034,6.364-0.057,6.035,0.114C5.706,0.287,5.5,0.628,5.5,1v40 c0,0.372,0.206,0.713,0.535,0.886C6.181,41.962,6.341,42,6.5,42c0.199,0,0.397-0.06,0.568-0.177L34.5,22.905V41c0,0.553,0.448,1,1,1 s1-0.447,1-1V1C36.5,0.447,36.052,0,35.5,0z"
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</svg> </button> </vli>
</vli>
<vli class="default__control vi__control__bottom">
<vli class="progress">
<vli class="progress__filled" style="flex-basis:0%"></vli>
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<vli style="float:left;position:absolute;right:5px;bottom:10px;height:initial"> <button type="button" class="default__button fullscreen" title="Fullscreen"> <svg class="" width="18px" height="18px"
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data-old_color="#ddd" data-original="#000000"></path>
</g>
</svg> </button> <button type="button" class="sound_off" id="vi_toggleVolVideoDiscovery" title="Enable Volume">
<vli class="vi__mutedVideo" style="padding:initial;background-color:transparent"> <svg class="icon-volume" fill="#ddd" height="19" viewBox="0 0 48 48" width="19"
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<path d="M0 0h48v48H0z" fill="none"></path>
</svg> </vli>
<vli class="vi__volVideo" style="padding:initial;background-color:transparent"> <svg class="icon-volume" fill="#ddd" height="19" viewBox="0 0 48 48" width="19"
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<vli style="float:left;position:absolute;left:5px;bottom:10px;height:initial"> <span class="vi__time vi__current_time">00:00</span><span class="vi__time vi__slash">/</span><span
class="vi__time vi__duration">00:00</span> </vli>
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<vli style="float: left; width: 33.3333%;">
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<span id="SiteMapPath1"><span><a title="Home" href="/default.aspx">Home</a></span><span> > </span><span>Operation Research calculators</span></span>
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<td align="center" colspan="2"> AtoZmath.com - Homework help (with all solution steps) </td>
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<b>Educational Level</b>
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<td> Secondary school, High school and College</td>
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<b>Program Purpose</b>
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<td> Provide step by step solutions of your problems using online calculators (online solvers) </td>
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<td> Your textbook, etc</td>
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<td align="center"> Operation Research Calculators <a href="//cbom.atozmath.com/Menu/CBomMenu.aspx"></a> (<a href="//cbom.atozmath.com/Menu/CBomMenu.aspx?he=e">examples</a>) </td>
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<b>1.</b> <b>Assignment problem</b>
<br>1.1 <a title="Assignment problem (Using Hungarian method)" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=hm2">Assignment problem (Using Hungarian method-2)</a>
<br>1.2 <a title="Assignment problem (Using Hungarian method)" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=hm">Assignment problem (Using Hungarian method-1)</a>
<br>2.1 <a title="Travelling salesman problem using hungarian method" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=tsh">Travelling salesman problem using hungarian method</a>
<br>2.2
<a title="Travelling salesman problem using branch and bound (penalty) method" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=tsp">Travelling salesman problem using branch and bound (penalty) method</a>
<br>2.3 <a title="Travelling salesman problem using branch and bound method" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=tsbb">Travelling salesman problem using branch and bound method</a>
<br>2.4 <a title="Travelling salesman problem using nearest neighbor method" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=tsnn">Travelling salesman problem using nearest neighbor method</a>
<br>2.5 <a title="Travelling salesman problem using diagonal completion method" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=tsdc">Travelling salesman problem using diagonal completion method</a>
<br>3. <a title="Crew assignment problem" href="//cbom.atozmath.com/CBOM/AssignmentCrew.aspx">Crew assignment problem</a>
<hr>
<b>2.</b> <b>Simplex method (Solve linear programming problem using)</b>
<br>0. <a title="Formulate linear programming model" href="//cbom.atozmath.com/example/CBOM/Simplex.aspx?q=lpm">Formulate linear programming model examples</a>
<br>1. <a title="Graphical method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=gm">Graphical method</a>
<br>2. <a title="Simplex method (BigM method)" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=sm">Simplex method (BigM method)</a>
<br>3. <a title="Two-Phase method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=tp">Two-Phase method</a>
<br>4. <a title="Primal to dual conversion" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=pd">Primal to dual conversion</a>
<br>5. <a title="Dual Simplex method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=ds">Dual Simplex method</a>
<br>6. <a title="Integer Simplex method " href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=is">Integer Simplex method (Gomory's cutting plane method)</a>
<br>7. <a title="Branch and Bound method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=bb">Branch and Bound method</a>
<br>8. <a title="0-1 Integer programming problem" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=01">0-1 Integer programming problem</a>
<br>9. <a title="Revised Simplex method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=rsm">Revised Simplex method</a>
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<b>3.</b> <b>Transportation Problem using</b>
<br>1. <a title="North-West Corner method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=nwcm">North-West corner method</a>
<br>2. <a title="Least Cost method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=lcm">Least cost method</a>
<br>3. <a title="Vogels Approximation method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=vam">Vogel's approximation method</a>
<br>4. <a title="Row minima method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=rm">Row minima method</a>
<br>5. <a title="Column minima method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=cm">Column minima method</a>
<br>6. <a title="Russells Approximation method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=ram">Russell's approximation method</a>
<br>7. <a title="Heuristic method-1" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=h1">Heuristic method-1</a>
<br>8. <a title="Heuristic method-2" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=h2">Heuristic method-2</a>
<br>9. <a title="Optimal solution using MODI method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=modi">Optimal solution using MODI method</a>
<br>10. <a title="Optimal solution using stepping stone method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=ss">Optimal solution using stepping stone method</a>
<hr>
<b>4.</b>
<b>PERT and CPM</b>
<br><b>1. Network diagram</b>
<br>1. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=1.1">Activity, Predecessors</a>
<br>2. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=2.1">Activity i-j</a>
<br>3. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=3.1">Activity i-j, Name of Activity</a>
<br>
<br><b>2. Critical path, Total float, Free float, Independent float</b>
<br>1. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=1.2">Activity, Predecessors, Duration</a>
<br>2. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=2.2">Activity i-j, Duration</a>
<br>3. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=3.2">Activity i-j, Name of Activity, Duration</a>
<br>
<br><b>3. Project scheduling with uncertain activity times (Optimistic, Most likely, Pessimistic)</b>
<br>1. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=1.3">Activity, Predecessors, to, tm, tp</a>
<br>2. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=2.3">Activity i-j, to, tm, tp</a>
<br>3. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=3.3">Activity i-j, Name of Activity, to, tm, tp</a>
<br>
<br><b>4. Project crashing to solve Time-Cost Trade-Off with fixed Indirect cost</b>
<br>1. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=1.4">Activity, Predecessors, Normal Time & Cost, Crash Time & Cost and Indirect Cost</a>
<br>2. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=2.4">Activity i-j, Normal Time & Cost, Crash Time & Cost and Indirect Cost</a>
<br>3. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=3.4">Activity i-j, Name of Activity, Normal Time & Cost, Crash Time & Cost and Indirect Cost</a>
<br>
<br><b>5. Project crashing to solve Time-Cost Trade-Off with varying Indirect cost</b>
<br>1. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=1.5">Activity, Predecessors, Normal Time & Cost, Crash Time & Cost and varying Indirect Cost</a>
<br>2. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=2.5">Activity i-j, Normal Time & Cost, Crash Time & Cost and varying Indirect Cost</a>
<br>3. <a title="" href="//cbom.atozmath.com/CBOM/PertCPM.aspx?q=3.5">Activity i-j, Name of Activity, Normal Time & Cost, Crash Time & Cost and varying Indirect Cost</a>
<br>
<hr>
<b>5.</b> <a title="Sequencing Problems" href="//cbom.atozmath.com/CBOM/SeqProb.aspx">Sequencing Problems</a>
<br>1. Processing n Jobs Through 2 Machines Problem <br>2. Processing n Jobs Through 3 Machines Problem <br>3. Processing n Jobs Through m Machines Problem <br>4. Processing 2 Jobs Through m Machines Problem
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<b>6.</b> <b>Replacement and Maintenance Models</b>
<br>1. Model-1 : Replacement policy for items whose running cost increases with time and value of money remains constant during a period <br>1.1
<a title="Replacement Model" href="//cbom.atozmath.com/CBOM/ReplaceModel11.aspx">Model-1.1</a>
<br>1.2 <a title="Replacement Model" href="//cbom.atozmath.com/CBOM/ReplaceModel12.aspx">Model-1.2</a>
<br>1.3 <a title="Replacement Model" href="//cbom.atozmath.com/CBOM/ReplaceModel13.aspx">Model-1.3</a>
<br>2. <a title="Replacement Model" href="//cbom.atozmath.com/CBOM/ReplaceModel21.aspx">Model-2</a> : Replacement policy for items whose running cost increases with time but value of money changes constant
rate during a period <br>3. <a title="Group Replacement Model" href="//cbom.atozmath.com/CBOM/ReplaceModel31.aspx">Model-3</a> : Group replacement policy
<hr>
<b>7.</b> <b>Game Theory</b>
<br>1. <a title="Saddle Point" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=saddlepoint">Saddle Point</a>
<br>2. <a title="Dominance method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=dominance">Dominance method</a>
<br>3. <a title="oddment method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=oddment">Oddment method</a>
<br>4. <a title="Algebraic method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=algebraic">Algebraic method</a>
<br>5. <a title="Calculus method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=calculus">Calculus method</a>
<br>6. <a title="Arithmetic method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=arithmetic">Arithmetic method</a>
<br>7. <a title="Matrix method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=matrix">Matrix method</a>
<br>8. <a title="2Xn Games" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=2xn">2Xn Games</a>
<br>9. <a title="Graphical method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=graph">Graphical method</a>
<br>10. <a title="LPP method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=lpp">LPP method</a>
<br>11. <a title="Bimatrix method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=bimatrix">Bimatrix method</a>
<hr>
<b>8.</b> <a title="Data envelopment analysis (DEA method)" href="//cbom.atozmath.com/CBOM/DEA.aspx">Data envelopment analysis (DEA method)</a>
</td>
</tr>
</tbody>
</table>
<hr>
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<br>
<table id="Table4" border="1" cellpadding="0" cellspacing="0" width="100%">
<tbody>
<tr class="menus1">
<td colspan="2" align="center"> Operation Research with example</td>
</tr>
<tr>
<td colspan="2" align="center" valign="top">
<table id="Table3" border="1" cellpadding="0" cellspacing="0" width="100%">
<tbody>
<tr class="menus1">
<td align="center" class="solutiontext">
<b>1.</b><a title="Assignment Problem" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=hm"></a> Assignment Problem
</td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>1.1</b><a title="Assignment Problem (Using Hungarian method)" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=hm">
Balanced Assignment Problem (Using Hungarian method)<br>
</a>
</td>
</tr>
<tr align="left">
<td> 1. A department has five employess with five jobs to be permormed. The time (in hours) each men will take to perform each job is given in the effectiveness matrix.<br>
<table class="tableHelp">
<tbody>
<tr align="center">
<td>
</td>
<td>
</td>
<td colspan="5"> Employees</td>
</tr>
<tr align="center">
<td>
</td>
<td>
</td>
<td> I</td>
<td> II</td>
<td> III</td>
<td> IV</td>
<td> V</td>
</tr>
<tr align="center">
<td rowspan="5"> Jobs</td>
<td> A</td>
<td> 10</td>
<td> 5</td>
<td> 113</td>
<td> 15</td>
<td> 16</td>
</tr>
<tr align="center">
<td> B</td>
<td> 3</td>
<td> 9</td>
<td> 18</td>
<td> 13</td>
<td> 6</td>
</tr>
<tr align="center">
<td> C</td>
<td> 10</td>
<td> 7</td>
<td> 2</td>
<td> 2</td>
<td> 2</td>
</tr>
<tr align="center">
<td> D</td>
<td> 7</td>
<td> 11</td>
<td> 9</td>
<td> 7</td>
<td> 12</td>
</tr>
<tr align="center">
<td> E</td>
<td> 7</td>
<td> 9</td>
<td> 10</td>
<td> 4</td>
<td> 12</td>
</tr>
</tbody>
</table> How should the jobs be allocated, one per employee, so as to minimize the total man-hours?<br>
<br>
<br>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>1.2</b> <a title="Assignment Problem (Using Hungarian method)" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=hm">
Unbalanced Assignment Problem (Using Hungarian method)<br>
</a>
</td>
</tr>
<tr align="left">
<td> 2. In the modification of a plant layout of a factory four new machines M1, M2, M3 and M4 are to be installed in a machine shop. There are five vacant places A, B, C, D and E available.
Because of limited space, machine M2 cannot be placed at C and M3 cannot be placed at A. The cost of locating a machine at a place (in hundred rupess) is as follows. <br>
<table class="tableHelp">
<tbody>
<tr align="center">
<td>
</td>
<td>
</td>
<td colspan="5"> Location</td>
</tr>
<tr align="center">
<td>
</td>
<td>
</td>
<td> A</td>
<td> B</td>
<td> C</td>
<td> D</td>
<td> E</td>
</tr>
<tr align="center">
<td rowspan="4"> Machine</td>
<td> M1</td>
<td> 9</td>
<td> 11</td>
<td> 15</td>
<td> 10</td>
<td> 11</td>
</tr>
<tr align="center">
<td> M2</td>
<td> 12</td>
<td> 9</td>
<td> --</td>
<td> 10</td>
<td> 9</td>
</tr>
<tr align="center">
<td> M3</td>
<td> --</td>
<td> 11</td>
<td> 14</td>
<td> 11</td>
<td> 7</td>
</tr>
<tr align="center">
<td> M4</td>
<td> 14</td>
<td> 8</td>
<td> 12</td>
<td> 7</td>
<td> 8</td>
</tr>
</tbody>
</table> Find the optimal assignment schedule.<br>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top"> </td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>2. Travelling salesman problem </b>
<br><b>2.1</b> <a title="Travelling salesman problem using hungarian method" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=tsh">using hungarian method</a>
<br><b>2.2</b> <a title="Travelling salesman problem using branch and bound (penalty) method" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=tsp">using branch and bound (penalty) method</a>
<br><b>2.3</b> <a title="Travelling salesman problem using branch and bound method" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=tsbb">using branch and bound method</a>
<br><b>2.4</b>
<a title="Travelling salesman problem using nearest neighbor method" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=tsnn">Travelling salesman problem using nearest neighbor method</a>
<br><b>2.5</b>
<a title="Travelling salesman problem using diagonal completion method" href="//cbom.atozmath.com/CBOM/Assignment.aspx?q=tsdc">Travelling salesman problem using diagonal completion method</a>
</td>
</tr>
<tr align="left">
<td> 1. A travelling salesman has to visit five cities. He wishes to start from a particular city, visit each city only once and then return to his starting point. The travelling cost of each
city from a particular city is given below. <br>
<table class="tableHelp">
<tbody>
<tr align="center">
<td></td>
<td></td>
<td colspan="5">To city</td>
</tr>
<tr align="center">
<td></td>
<td></td>
<td>A</td>
<td>B</td>
<td>C</td>
<td>D</td>
<td>E</td>
</tr>
<tr align="center">
<td rowspan="5">From city</td>
<td>A</td>
<td>x</td>
<td>2</td>
<td>5</td>
<td>7</td>
<td>1</td>
</tr>
<tr align="center">
<td>B</td>
<td>6</td>
<td>x</td>
<td>3</td>
<td>8</td>
<td>2</td>
</tr>
<tr align="center">
<td>C</td>
<td>8</td>
<td>7</td>
<td>x</td>
<td>4</td>
<td>7</td>
</tr>
<tr align="center">
<td>D</td>
<td>12</td>
<td>4</td>
<td>6</td>
<td>x</td>
<td>5</td>
</tr>
<tr align="center">
<td>E</td>
<td>1</td>
<td>3</td>
<td>2</td>
<td>8</td>
<td>x</td>
</tr>
</tbody>
</table> How should the jobs be allocated, one per employee, so as to minimize the total man-hours?
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>3</b> <a title="Crew assignment problem" href="//cbom.atozmath.com/CBOM/AssignmentCrew.aspx">
Crew assignment problem<br>
</a>
</td>
</tr>
<tr align="left">
<td> 1. Best-ride airlines that operates seven days a week has the following time-table. <br>
<table class="tableHelp">
<tbody>
<tr>
<td align="center"> Delhi - Mumbai </td>
<td>
</td>
<td align="center"> Mumbai - Delhi </td>
</tr>
<tr>
<td align="center">
<table class="tableHelp">
<tbody>
<tr align="center">
<td> Flight No</td>
<td> Departure</td>
<td> Arrival</td>
</tr>
<tr align="center">
<td> 1</td>
<td> 7.00</td>
<td> 8.00</td>
</tr>
<tr align="center">
<td> 2</td>
<td> 8.00</td>
<td> 9.00</td>
</tr>
<tr align="center">
<td> 3</td>
<td> 13.00</td>
<td> 14.00</td>
</tr>
<tr align="center">
<td> 4</td>
<td> 18.00</td>
<td> 19.00</td>
</tr>
</tbody>
</table>
</td>
<td>
</td>
<td align="center">
<table id="Table10" class="table0" width="100%" border="1">
<tbody>
<tr align="center">
<td> Flight No</td>
<td> Departure</td>
<td> Arrival</td>
</tr>
<tr align="center">
<td> 101</td>
<td> 8.00</td>
<td> 9.00</td>
</tr>
<tr align="center">
<td> 102</td>
<td> 9.00</td>
<td> 10.00</td>
</tr>
<tr align="center">
<td> 103</td>
<td> 12.00</td>
<td> 13.00</td>
</tr>
<tr align="center">
<td> 104</td>
<td> 17.00</td>
<td> 18.00</td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
<br> Crews must have a minimum layover of 5 hours between flights. Obtain the pairing of flights that minimizes layover time away from home. For any given pairing, the crew will be based at
the city that results in the smaller layover. For each pair also mention the city where crew should be based. <br>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top" colspan="2">
<table id="Table1" border="1" cellpadding="0" cellspacing="0" width="100%">
<tbody>
<tr class="menus1">
<td colspan="2" align="center">
<b>2.</b> Simplex method<br> Solve the following LP problem by using <br>
<a title="Simplex method (BigM method)" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=sm">
1. Simplex method (BigM method)
</a>
<br>
<a title="Two-Phase method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=tp">
2. Two-Phase method
</a>
<br>
<a title="Graphical method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=gm">
3. Graphical method
</a>
<br>
<a title="Primal to dual conversion" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=pd">
4. Primal to dual conversion
</a>
<br>
<a title="Dual Simplex method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=ds">
5. Dual Simplex method
</a>
<br>
<a title="Integer Simplex method " href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=is">
6. Integer Simplex method (Gomory's cutting plane method)
</a>
<br>
<a title="Branch and Bound method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=bb">
7. Branch and Bound method
</a>
<br>
<a title="0-1 Integer programming problem" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=01">
8. 0-1 Integer programming problem
</a>
<br>
<a title="Revised Simplex method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=rsm">
9. Revised Simplex method
</a>
<br>
</td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>2.1</b>
<a title="Simplex method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=sm">
Simplex method</a>
</td>
</tr>
<tr align="left">
<td> 1. Use the simplex method to solve the following LP problem.<br> Maximize Z = 3x1 + 5x2 + 4x3<br> subject to the constraints<br> 2x1 + 3x2 ≤ 8<br> 2x2 + 5x3 ≤ 10<br> 3x1 + 2x2 + 4x3 ≤
15<br> and x1, x2, x3 ≥ 0<br>
<br> 2. Use the simplex method to solve the following LP problem.<br> Maximize Z = 4x1 + 3x2<br> subject to the constraints<br> 2x1 + x2 ≤ 1000<br> x1 + x2 ≤ 800<br> x1 ≤ 400<br> x2 ≤ 700<br>
and x1, x2 ≥ 0<br>
</td>
</tr>
</tbody>
</table>
</td>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>2.2</b>
<a title="BigM method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=sm">
BigM method</a>
</td>
</tr>
<tr align="left">
<td> 1. Use the penalty (Big - M) method to solve the following LP problem.<br> Minimize Z = 5x1 + 3x2<br> subject to the constraints<br> 2x1 + 4x2 ≤ 12<br> 2x1 + 2x2 = 10<br> 5x1 + 2x2 ≥ 10<br>
and x1, x2 ≥ 0<br>
<br> 2. Use the penalty (Big - M) method to solve the following LP problem.<br> Minimize Z = x1 + 2x2 + 3x3 - x4<br> subject to the constraints<br> x1 + 2x2 + 3x3 = 15<br> 2x1 + x2 + 5x3 =
20<br> x1 + 2x2 + x3 + x4 = 10<br> and x1, x2, x3, x4 ≥ 0
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>2.3</b>
<a title="Two-Phase method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=tp">
Two-Phase method</a>
</td>
</tr>
<tr align="left">
<td> 1. Solve the following LP problem by using the Two-Phase method.<br> Minimize Z = x1 + x2<br> subject to the constraints<br> 2x1 + 4x2 ≥ 4<br> x1 + 7x2 ≥ 7<br> and x1, x2 ≥ 0<br>
<br> 2. Solve the following LP problem by using the Two-Phase method.<br> Minimize Z = x1 - 2x2 - 3x3<br> subject to the constraints<br> -2x1 + 3x2 + 3x3 = 2<br> 2x1 + 3x2 + 4x3 = 1<br> and
x1, x2, x3 ≥ 0
</td>
</tr>
</tbody>
</table>
</td>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>2.4</b>
<a title="Dual Simplex method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=ds">
Dual Simplex method</a>
</td>
</tr>
<tr align="left">
<td> 1. Solve the following LP problem by using the Two-Phase method.<br> Minimize Z = x1 + x2<br> subject to the constraints<br> 2x1 + 4x2 ≥ 4<br> x1 + 7x2 ≥ 7<br> and x1, x2 ≥ 0<br>
<br> 2. Solve the following LP problem by using the Two-Phase method.<br> Minimize Z = x1 - 2x2 - 3x3<br> subject to the constraints<br> -2x1 + 3x2 + 3x3 = 2<br> 2x1 + 3x2 + 4x3 = 1<br> and
x1, x2, x3 ≥ 0
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>2.5</b>
<a title="Gomorys Integer Cutting method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=is">
Gomorys Integer Cutting method</a>
</td>
</tr>
<tr align="left">
<td> 1. Solve the following integer programming problem using Gomory's cutting plane algorithm.<br> Maximize Z = x1 + x2<br> subject to the constraints<br> 3x1 + 2x2 ≤ 5<br> x2 ≤ 2<br> and x1,
x2 ≥ 0 and are integers.<br>
<br> 2. Solve the following integer programming problem using Gomory's cutting plane algorithm.<br> Maximize Z = 2x1 + 20x2 - 10x3<br> subject to the constraints<br> 2x1 + 20x2 + 4x3 ≤ 15<br>
6x1 + 20x2 + 4x3 ≤ 20<br> and x1, x2, x3 ≥ 0 and are integers.
</td>
</tr>
</tbody>
</table>
</td>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>2.6</b>
<a title="Graphical method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=gm">
Graphical method</a>
</td>
</tr>
<tr align="left">
<td> 1. Use graphical method to solve following LP problem.<br> Maximize Z = x1 + x2<br> subject to the constraints<br> 3x1 + 2x2 ≤ 5<br> x2 ≤ 2<br> and x1, x2 ≥ 0<br>
<br> 2. Use graphical method to solve following LP problem.<br> Maximize Z = 2x1 + x2<br> subject to the constraints<br> x1 + 2x2 ≤ 10<br> x1 + x2 ≤ 6<br> x1 - x2 ≤ 2<br> x1 - 2x2 ≤ 1<br> and
x1, x2 ≥ 0<br>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>2.7</b>
<a title="Primal to dual conversion" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=pd">
Primal to dual conversion</a>
</td>
</tr>
<tr align="left">
<td> 1. Write the dual to the following LP problem.<br> Maximize Z = x1 - x2 + 3x3<br> subject to the constraints<br> x1 + x2 + x3 ≤ 10<br> 2x1 - x2 - x3 ≤ 2<br> 2x1 - 2x2 - 3x3 ≤ 6<br> and x1,
x2, x3 ≥ 0<br>
<br> 2. Write the dual to the following LP problem.<br> Minimize Z = 3x1 - 2x2 + 4x3<br> subject to the constraints<br> 3x1 + 5x2 + 4x3 ≥ 7<br> 6x1 + x2 + 3x3 ≥ 4<br> 7x1 - 2x2 - x3 ≤ 10<br>
x1 - 2x2 + 5x3 ≥ 3<br> 4x1 + 7x2 - 2x3 ≥ 2<br> and x1, x2, x3 ≥ 0
</td>
</tr>
</tbody>
</table>
</td>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>2.8</b>
<a title="Branch and Bound method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=bb">
Branch and Bound method</a>
</td>
</tr>
<tr align="left">
<td> 1. Solve the following LP problem by using Branch and Bound method<br> Max Z = 7x1 + 9x2<br> subject to <br> -x1 + 3x2 ≤ 6<br> 7x1 + x2 ≤ 35<br> x2 ≤ 7<br> and x1,x2 ≥ 0 <br>
<br> 2. Solve the following LP problem by using Branch and Bound method<br> Max Z = 3x1 + 5x2<br> subject to <br> 2x1 + 4x2 ≤ 25<br> x1 ≤ 8<br> 2x2 ≤ 10<br> and x1,x2 ≥ 0 <br>
<br>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>2.9</b>
<a title="0-1 Integer programming problem" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=01">
0-1 Integer programming problem</a>
</td>
</tr>
<tr align="left">
<td> 1. Solve LP using zero-one Integer programming problem method<br> Max Z = 300x1 + 90x2 + 400x3 + 150x4<br> subject to <br> 35000x1 + 10000x2 + 25000x3 + 90000x4 ≤ 120000<br> 4x1 + 2x2 + 7x3
+ 3x4 ≤ 12<br> x1 + x2 ≤ 1<br> and x1,x2,x3,x4 ≥ 0 <br>
<br> 2. Solve LP using 0-1 Integer programming problem method<br> MAX Z = 650x1 + 700x2 + 225x3 + 250x4<br> subject to<br> 700x1 + 850x2 + 300x3 + 350x4 ≤ 1200<br> 550x1 + 550x2 + 150x3 +
200x4 ≤ 700<br> 400x1 + 350x2 + 100x3 ≤ 400<br> x1 + x2 ≥ 1<br> -x3 + x4 ≤ 1<br> and x1,x2,x3,x4 ≥ 0 <br>
</td>
</tr>
</tbody>
</table>
</td>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>2.9</b>
<a title="Revised Simplex method" href="//cbom.atozmath.com/CBOM/Simplex.aspx?q=rsm">
Revised Simplex method</a>
</td>
</tr>
<tr align="left">
<td> 1. Solve the following LP problem by using Revised Simplex method<br> MAX Z = 3x1 + 5x2<br> subject to <br> x1 ≤ 4<br> x2 ≤ 6<br> 3x1 + 2x2 ≤ 18<br> and x1,x2 ≥ 0 <br>
<br> 2. Solve the following LP problem by using Revised Simplex method<br> MAX Z = 2x1 + x2<br> subject to<br> 3x1 + 4x2 ≤ 6<br> 6x1 + x2 ≤ 3<br> and x1,x2 ≥ 0 <br>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top" colspan="2">
<table id="Table2" border="1" cellpadding="0" cellspacing="0" width="100%">
<tbody>
<tr class="menus1">
<td align="center">
<b>3.</b> Transportation Problem using <br>
<a title="North-West Corner method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=nwcm" "="">1. North-West corner method</a>
<br><a title="Least Cost method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=lcm">2. Least cost method</a>
<br><a title="Vogels Approximation method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=vam">3. Vogel's approximation method</a>
<br><a title="Row minima method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=rm">4. Row minima method</a>
<br><a title="Column minima method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=cm">5. Column minima method</a>
<br><a title="Russells Approximation method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=ram">6. Russell's approximation method</a>
<br><a title="Heuristic method-1" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=h1">7. Heuristic method-1</a>
<br><a title="Heuristic method-2" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=h2">8. Heuristic method-1</a>
<br><a title="Vogels Approximation method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=modi">9. optimal solution by MODI method </a>
<br><a title="Optimal solution using stepping stone method" href="//cbom.atozmath.com/CBOM/Transportation.aspx?q=ss">10. Optimal solution using stepping stone method</a>
</td>
</tr>
<tr>
<td align="center" valign="top" width="50%">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
</td>
</tr>
<tr align="left">
<td> 1. A Company has 3 production facilities S1, S2 and S3 with production capacity of 7, 9 and 18 units (in 100's) per week of a product, respectively. These units are tobe shipped to 4
warehouses D1, D2, D3 and D4 with requirement of 5,6,7 and 14 units (in 100's) per week, respectively. The transportation costs (in rupees) per unit between factories to warehouses are given
in the table below. <br>
<table class="tableHelp">
<tbody>
<tr align="center">
<td>
</td>
<td> D<sub>1</sub></td>
<td> D<sub>2</sub></td>
<td> D<sub>3</sub></td>
<td> D<sub>4</sub></td>
<td> Capacity</td>
</tr>
<tr align="center">
<td> S<sub>1</sub></td>
<td> 19</td>
<td> 30</td>
<td> 50</td>
<td> 10</td>
<td> 7</td>
</tr>
<tr align="center">
<td> S<sub>2</sub></td>
<td> 70</td>
<td> 30</td>
<td> 40</td>
<td> 60</td>
<td> 9</td>
</tr>
<tr align="center">
<td> S<sub>3</sub></td>
<td> 40</td>
<td> 8</td>
<td> 70</td>
<td> 20</td>
<td> 18</td>
</tr>
<tr align="center">
<td> Demand</td>
<td> 5</td>
<td> 8</td>
<td> 7</td>
<td> 14</td>
<td> 34</td>
</tr>
</tbody>
</table>
<br> Find initial basic feasible solution for given problem by using<br> (a) North-West corner method<br> (b) Least cost method<br> (c) Vogel's approximation method<br> (d) obtain an optimal
solution by MODI method<br> if the object is to minimize the total transportation cost. <br>
<br>
<br> 2. Find an initial basic feasible solution for given transportation problem by using<br> (a) North-West corner method<br> (b) Least cost method<br> (c) Vogel's approximation method<br>
<table class="tableHelp">
<tbody>
<tr align="center">
<td>
</td>
<td> D<sub>1</sub></td>
<td> D<sub>2</sub></td>
<td> D<sub>3</sub></td>
<td> D<sub>4</sub></td>
<td> Supply</td>
</tr>
<tr align="center">
<td> S<sub>1</sub></td>
<td> 11</td>
<td> 13</td>
<td> 17</td>
<td> 14</td>
<td> 250</td>
</tr>
<tr align="center">
<td> S<sub>2</sub></td>
<td> 16</td>
<td> 18</td>
<td> 14</td>
<td> 10</td>
<td> 300</td>
</tr>
<tr align="center">
<td> S<sub>3</sub></td>
<td> 21</td>
<td> 24</td>
<td> 13</td>
<td> 10</td>
<td> 400</td>
</tr>
<tr align="center">
<td> Demand</td>
<td> 200</td>
<td> 225</td>
<td> 275</td>
<td> 250</td>
<td>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
</td>
</tr>
<tr align="left">
<td> 3. A company has factories at F1, F2 and F3 which supply to warehouses at W1, W2 and W3. Weekly factory capacities are 200, 160 and 90 units, respectively. Weekly warehouse requiremnet are
180, 120 and 150 units, respectively. Unit shipping costs (in rupess) are as follows:<br>
<table class="tableHelp">
<tbody>
<tr align="center">
<td>
</td>
<td> W<sub>1</sub></td>
<td> W<sub>2</sub></td>
<td> W<sub>3</sub></td>
<td> Supply</td>
</tr>
<tr align="center">
<td> F<sub>1</sub></td>
<td> 16</td>
<td> 20</td>
<td> 12</td>
<td> 200</td>
</tr>
<tr align="center">
<td> F<sub>2</sub></td>
<td> 14</td>
<td> 8</td>
<td> 18</td>
<td> 160</td>
</tr>
<tr align="center">
<td> F<sub>3</sub></td>
<td> 26</td>
<td> 24</td>
<td> 16</td>
<td> 90</td>
</tr>
<tr align="center">
<td> Demand</td>
<td> 180</td>
<td> 120</td>
<td> 150</td>
<td> 450</td>
</tr>
</tbody>
</table>
<br> Determine the optimal distribution for this company to minimize total shipping cost.<br>
<br>
<br> 4. Find an initial basic feasible solution for given transportation problem by using<br> (a) North-West corner method<br> (b) Least cost method<br> (c) Vogel's approximation method<br>
<table class="tableHelp">
<tbody>
<tr align="center">
<td>
</td>
<td> P</td>
<td> Q</td>
<td> R</td>
<td> S</td>
<td> Supply</td>
</tr>
<tr align="center">
<td> A</td>
<td> 6</td>
<td> 3</td>
<td> 5</td>
<td> 4</td>
<td> 22</td>
</tr>
<tr align="center">
<td> B</td>
<td> 5</td>
<td> 9</td>
<td> 2</td>
<td> 7</td>
<td> 15</td>
</tr>
<tr align="center">
<td> C</td>
<td> 5</td>
<td> 7</td>
<td> 8</td>
<td> 6</td>
<td> 8</td>
</tr>
<tr align="center">
<td> Demand</td>
<td> 7</td>
<td> 12</td>
<td> 17</td>
<td> 9</td>
<td> 45</td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top" colspan="2">
<table id="Table5" border="1" cellpadding="0" cellspacing="0" width="100%">
<tbody>
<tr class="menus1">
<td align="center"> 4. <a title="PERT and CPM" href="//cbom.atozmath.com/CBOM/PertCPM.aspx">PERT and CPM</a>
</td>
</tr>
<tr>
<td align="center" valign="top" width="50%">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
</td>
</tr>
<tr align="left">
<td> 1. An assembly is to be made from two parts X and Y. Both parts must be turned on a lathe Y must be polished where as X need not be polished. The sequence of acitivities, together with
their predecessors, is given below <table class="tableHelp">
<tbody>
<tr align="center">
<td>Activity</td>
<td>Description</td>
<td>Predecessor Activity</td>
</tr>
<tr align="center">
<td>A</td>
<td>Open work order</td>
<td>-</td>
</tr>
<tr align="center">
<td> B</td>
<td> Get material for X</td>
<td> A</td>
</tr>
<tr align="center">
<td> C</td>
<td> Get material for Y</td>
<td> A</td>
</tr>
<tr align="center">
<td> D</td>
<td> Turn X on lathe</td>
<td> B</td>
</tr>
<tr align="center">
<td> E</td>
<td> Turn Y on lathe</td>
<td> B,C</td>
</tr>
<tr align="center">
<td> F</td>
<td> Polish Y</td>
<td> E</td>
</tr>
<tr align="center">
<td> G</td>
<td> Assemble X and Y</td>
<td> D,F</td>
</tr>
<tr align="center">
<td> H</td>
<td> Pack</td>
<td> G</td>
</tr>
</tbody>
</table> Draw a network diagram of activities for the project.<br>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
</td>
</tr>
<tr align="left">
<td> 2. An established company has decided to add a new product to its line. It will buy the product from a manufacturing concern, package it, and sell it to a number of distributors that have
been selected on a geographical basis. Market research has already indicated the volume expected and the size of sales force required. The steps shown in the following table are to be planned.
<table class="tableHelp">
<tbody>
<tr align="center">
<td>Activity</td>
<td>Description</td>
<td>Predecessor Activity</td>
<td> Duration (days)</td>
</tr>
<tr align="center">
<td>A</td>
<td> Organize sales office</td>
<td>-</td>
<td> 6</td>
</tr>
<tr align="center">
<td> B</td>
<td> Hire salesman</td>
<td> A</td>
<td> 4</td>
</tr>
<tr align="center">
<td> C</td>
<td> Train salesman</td>
<td> B</td>
<td> 7</td>
</tr>
<tr align="center">
<td> D</td>
<td> Select advertising agency</td>
<td> A</td>
<td> 2</td>
</tr>
<tr align="center">
<td> E</td>
<td> Plan advertising campaign</td>
<td> D</td>
<td> 4</td>
</tr>
<tr align="center">
<td> F</td>
<td> Conduct advertising campaign</td>
<td> E</td>
<td> 10</td>
</tr>
<tr align="center">
<td> G</td>
<td> Design package</td>
<td> -</td>
<td> 2</td>
</tr>
<tr align="center">
<td> H</td>
<td> Setup packaging campaign</td>
<td> G</td>
<td> 10</td>
</tr>
<tr align="center">
<td> I</td>
<td> Package initial stocks</td>
<td> J,H</td>
<td> 6</td>
</tr>
<tr align="center">
<td> J</td>
<td> Order stock from manufacturer</td>
<td> -</td>
<td> 13</td>
</tr>
<tr align="center">
<td> K</td>
<td> Select distributors</td>
<td> A</td>
<td> 9</td>
</tr>
<tr align="center">
<td> L</td>
<td> Sell to distributors</td>
<td> C,K</td>
<td> 3</td>
</tr>
<tr align="center">
<td> M</td>
<td> Ship stocks to distributors</td>
<td> I,L</td>
<td> 5</td>
</tr>
</tbody>
</table> (a) Draw an arrow diagram for the project.<br> (b) Indicate the criticla path.<br> (c) For each non-critical activity, find the total and free float.<br>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top" colspan="2">
<table id="Table6" border="1" cellpadding="0" cellspacing="0" width="100%">
<tbody>
<tr class="menus1">
<td align="center"> 5. <a title="Sequencing Problems" href="//cbom.atozmath.com/CBOM/SeqProb.aspx">Sequencing Problems</a>
</td>
</tr>
<tr>
<td align="center" valign="top" width="50%">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
</td>
</tr>
<tr align="left">
<td> 1. There are seven jobs, each of which has to go through the machines A and B in the order AB. Processing times in hours are as follows. <table class="tableHelp">
<tbody>
<tr align="center">
<td> Job</td>
<td>1</td>
<td> 2</td>
<td> 3</td>
<td> 4</td>
<td> 5</td>
<td> 6</td>
<td> 7</td>
</tr>
<tr align="center">
<td> Machine A</td>
<td> 3</td>
<td> 12</td>
<td> 15</td>
<td> 6</td>
<td> 10</td>
<td> 11</td>
<td> 9</td>
</tr>
<tr align="center">
<td> Machine B</td>
<td> 8</td>
<td> 10</td>
<td> 10</td>
<td> 6</td>
<td> 12</td>
<td> 1</td>
<td> 3</td>
</tr>
</tbody>
</table> Decide a sequence of these jobs that will minimize the total elapsed time T. Also find T and idle time for machines A and B.<br>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
</td>
</tr>
<tr align="left">
<td> 2. Find the sequence that minimizes the total time required in performing the following job on three machines in the order ABC. Processing times (in hours) are given in the following table.
<table class="tableHelp">
<tbody>
<tr align="center">
<td> Job</td>
<td>1</td>
<td> 2</td>
<td> 3</td>
<td> 4</td>
<td> 5</td>
</tr>
<tr align="center">
<td> Machine A</td>
<td> 8</td>
<td> 10</td>
<td> 6</td>
<td> 7</td>
<td> 11</td>
</tr>
<tr align="center">
<td> Machine B</td>
<td> 5</td>
<td> 6</td>
<td> 2</td>
<td> 3</td>
<td> 4</td>
</tr>
<tr align="center">
<td> Machine C</td>
<td> 4</td>
<td> 9</td>
<td> 8</td>
<td> 6</td>
<td> 5</td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top" colspan="2">
<table id="Table7" border="1" cellpadding="0" cellspacing="0" width="100%">
<tbody>
<tr class="menus1">
<td colspan="2" align="center"> 6. <b>Replacement and Maintenance Models</b>
</td>
</tr>
<tr>
<td colspan="2" align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<a title="Replacement Model" href="//cbom.atozmath.com/CBOM/ReplaceModel11.aspx">Model-1.1</a>
</td>
</tr>
<tr align="left">
<td> 1. A firm is considering the replacement of a machine, whose cost price is Rs 12,200 and its scrap value is Rs 200. From experience the running (maintenance and operating) costs are found
to be as follows: <br>
<table border="1" cellspacing="0" cellpadding="4">
<tbody>
<tr align="center">
<th nowrap="">Year</th>
<td nowrap="">1</td>
<td nowrap="">2</td>
<td nowrap="">3</td>
<td nowrap="">4</td>
<td nowrap="">5</td>
<td nowrap="">6</td>
<td nowrap="">7</td>
<td nowrap="">8</td>
</tr>
<tr align="center">
<th nowrap="">Running Cost</th>
<td nowrap="">200</td>
<td nowrap="">500</td>
<td nowrap="">800</td>
<td nowrap="">1,200</td>
<td nowrap="">1,800</td>
<td nowrap="">2,500</td>
<td nowrap="">3,200</td>
<td nowrap="">4,000</td>
</tr>
</tbody>
</table>
<br>When should the machine be replaced?
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td colspan="2" align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<a title="Replacement Model" href="//cbom.atozmath.com/CBOM/ReplaceModel12.aspx">Model-1.2</a>
</td>
</tr>
<tr align="left">
<td> 1. The data collected in running a machine, the cost of which is Rs 60,000 are given below: <br>
<table border="1" cellspacing="0" cellpadding="4">
<tbody>
<tr align="center">
<th nowrap="">Year</th>
<td nowrap="">1</td>
<td nowrap="">2</td>
<td nowrap="">3</td>
<td nowrap="">4</td>
<td nowrap="">5</td>
</tr>
<tr align="center">
<th nowrap="">Resale Value</th>
<td nowrap="">42,000</td>
<td nowrap="">30,000</td>
<td nowrap="">20,400</td>
<td nowrap="">14,400</td>
<td nowrap="">9,650</td>
</tr>
<tr align="center">
<th nowrap="">Cost of spares</th>
<td nowrap="">4,000</td>
<td nowrap="">4,270</td>
<td nowrap="">4,880</td>
<td nowrap="">5,700</td>
<td nowrap="">6,800</td>
</tr>
<tr align="center">
<th nowrap="">Cost of labour</th>
<td nowrap="">14,000</td>
<td nowrap="">16,000</td>
<td nowrap="">18,000</td>
<td nowrap="">21,000</td>
<td nowrap="">25,000</td>
</tr>
</tbody>
</table>
<br> Determine the optimum period for replacement of the machine.
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td colspan="2" align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<a title="Replacement Model" href="//cbom.atozmath.com/CBOM/ReplaceModel13.aspx">Model-1.3</a>
</td>
</tr>
<tr align="left">
<td> 1. Machine A costs Rs 45,000 and its operating costs are estimated to be Rs 1,000 for the first year increasing by Rs 10,000 per year in the second and subsequent years. Machine B costs Rs
50,000 and operating costs are Rs 2,000 for the first year, increasing by Rs 4,000 in the second and subsequent years. If at present we have a machine of type A, should we replace it with B?
if so when? Assume that both machines have no resale value and their future costs are not discounted. </td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td colspan="2" align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<a title="Replacement Model" href="//cbom.atozmath.com/CBOM/ReplaceModel21.aspx">Model-2</a>
<br>Replacement policy for items whose running cost increases with time but value of money changes constant rate during a period
</td>
</tr>
<tr align="left">
<td> 1. An engineering company is offered a material handling equipment A. It is priced at Rs 60,000 includeing cost of installation. The costs for operation and maintenance are estimated to be
Rs 10,000 for each of the first five years, increasing every year by Rs 3,000 in the sixth and subsequent years. The company expects a return of 10 percent on all its investment. What is the
optimal replacement period? <br>
<table border="1" cellspacing="0" cellpadding="4">
<tbody>
<tr align="center">
<th nowrap="">Year</th>
<td nowrap="">1</td>
<td nowrap="">2</td>
<td nowrap="">3</td>
<td nowrap="">4</td>
<td nowrap="">5</td>
<td nowrap="">6</td>
<td nowrap="">7</td>
</tr>
<tr align="center">
<th nowrap="">Running Cost</th>
<td nowrap="">10,000</td>
<td nowrap="">10,000</td>
<td nowrap="">10,000</td>
<td nowrap="">10,000</td>
<td nowrap="">10,000</td>
<td nowrap="">13,000</td>
<td nowrap="">16,000</td>
</tr>
</tbody>
</table>
<br>
<br> 2. A company is buying mini computers. It costs Rs 5 lakh, and its running and maintenance costs are Rs 60,000 for each of the first five years, increasing by Rs 20,000 per year in the
sixth and subsequent years. If the money is worth 10 percent per year, What is the optimal replacement period?
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td colspan="2" align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<a title="Group Replacement Model" href="//cbom.atozmath.com/CBOM/ReplaceModel31.aspx">Model-3</a> <br> Group replacement policy
</td>
</tr>
<tr align="left">
<td> 1. A computer contains 10,000 resistors. When any resistor fails, it is replaced. The cost of replacing a resistor individually is Rs 1 only. If all the resistors are replaced at the same
time, the cost per resistor would be reduced to 35 paise. The percentage of surviving resistors say S(t) at the end of month t and the probability of failure P(t) during the month t are as
follows: <br>
<table border="1" cellspacing="0" cellpadding="4">
<tbody>
<tr align="center">
<th nowrap="">t</th>
<td nowrap="">0</td>
<td nowrap="">1</td>
<td nowrap="">2</td>
<td nowrap="">3</td>
<td nowrap="">4</td>
<td nowrap="">5</td>
<td nowrap="">6</td>
</tr>
<tr align="center">
<th nowrap="">P(t)</th>
<td nowrap="">0</td>
<td nowrap="">0.03</td>
<td nowrap="">0.07</td>
<td nowrap="">0.20</td>
<td nowrap="">0.40</td>
<td nowrap="">0.15</td>
<td nowrap="">0.15</td>
</tr>
</tbody>
</table> What is the optimal replacement plan? <br>
<br> 2. The following mortality rates have been observed for a certain type of fuse: <br>
<table border="1" cellspacing="0" cellpadding="4">
<tbody>
<tr align="center">
<th nowrap="">t</th>
<td nowrap="">0</td>
<td nowrap="">1</td>
<td nowrap="">2</td>
<td nowrap="">3</td>
<td nowrap="">4</td>
<td nowrap="">5</td>
</tr>
<tr align="center">
<th nowrap="">P(t)</th>
<td nowrap="">0</td>
<td nowrap="">0.05</td>
<td nowrap="">0.10</td>
<td nowrap="">0.20</td>
<td nowrap="">0.40</td>
<td nowrap="">0.25</td>
</tr>
</tbody>
</table> There are 1,000 fuses in use and it costs Rs 5 to replace an individual fuse. If all fuses were replaced simultaneously it would cost Rs 1.25 per fuse. It is proposed to replace all
fuses at fixed intervals of time, whether or not they have burnt out, and to contiune replacing burnt out fuses as they fail. At what time intervals should the group replacement be made? Also
prove that this optimal policy is superior to the straight forward policy of replacing each fuse only when it fails.
</td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top" colspan="2">
<table id="Table8" border="1" cellpadding="0" cellspacing="0" width="100%">
<tbody>
<tr class="menus1">
<td colspan="2" align="center">
<b>7.</b> <b>Game Theory</b>
<br> <a title="Saddle Point" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=saddlepoint">
1. Saddle Point
</a>
<br> <a title="Dominance method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=dominance">
2. Dominance method
</a>
<br> <a title="Oddment method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=oddment">
3. Oddment method
</a>
<br> <a title="Algebraic method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=algebraic">
4. Algebraic method
</a>
<br> <a title="Calculus method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=calculus">
5. Calculus method
</a>
<br> <a title="Arithmetic method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=arithmetic">
6. Arithmetic method
</a>
<br> <a title="Matrix method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=matrix">
7. Matrix method
</a>
<br> <a title="2Xn Games" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=2xn">
8. 2Xn Games
</a>
<br> <a title="Graphical method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=graph">
9. Graphical method
</a>
<br> <a title="LPP method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=lpp">
10. LPP method
</a>
<br> <a title="Bimatrix method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=bimatrix">
11. Bimatrix method
</a>
</td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>7.1</b>
<a title="Saddle Point" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=saddlepoint">Saddle Point</a>
</td>
</tr>
<tr align="left">
<td> 1. For the game with payoff matrix <table border="0" cellspacing="0" cellpadding="5" style="border-collapse: collapse">
<tbody>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap="" colspan="3">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-1-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-1"><span class="MJXp-mstyle" id="MJXp-Span-2"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-3">B</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-1">B</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-2-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-4"><span class="MJXp-mstyle" id="MJXp-Span-5"><span class="MJXp-msub" id="MJXp-Span-6"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-7"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-8" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-2">B_1</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-3-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-9"><span class="MJXp-mstyle" id="MJXp-Span-10"><span class="MJXp-msub" id="MJXp-Span-11"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-12"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-13" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-3">B_2</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-4-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-14"><span class="MJXp-mstyle" id="MJXp-Span-15"><span class="MJXp-msub" id="MJXp-Span-16"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-17"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-18" style="vertical-align: -0.4em;">3</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-4">B_3</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap="" rowspan="2">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-5-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-19"><span class="MJXp-mstyle" id="MJXp-Span-20"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-21">A</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-5">A</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-6-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-22"><span class="MJXp-mstyle" id="MJXp-Span-23"><span class="MJXp-msub" id="MJXp-Span-24"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-25"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-26" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-6">A_1</script>
</td>
<td class="MatrixTL"></td>
<td nowrap=""> -1 </td>
<td nowrap=""> 2 </td>
<td nowrap=""> -2 </td>
<td class="MatrixTR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-7-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-27"><span class="MJXp-mstyle" id="MJXp-Span-28"><span class="MJXp-msub" id="MJXp-Span-29"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-30"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-31" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-7">A_2</script>
</td>
<td class="MatrixBL"></td>
<td nowrap=""> 6 </td>
<td nowrap=""> 4 </td>
<td nowrap=""> -6 </td>
<td class="MatrixBR"></td>
</tr>
</tbody>
</table> determine the best strategies for players A and B. Also determine the value of game. Is this game saddle point? </td>
</tr>
</tbody>
</table>
</td>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>7.2</b>
<a title="Dominance method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=dominance">Dominance method</a>
</td>
</tr>
<tr align="left">
<td> 1. Dominance Example <table border="0" cellspacing="0" cellpadding="5" style="border-collapse: collapse">
<tbody>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap="" colspan="4">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-8-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-32"><span class="MJXp-mstyle" id="MJXp-Span-33"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-34">B</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-8">B</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-9-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-35"><span class="MJXp-mstyle" id="MJXp-Span-36"><span class="MJXp-msub" id="MJXp-Span-37"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-38"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-39" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-9">B_1</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-10-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-40"><span class="MJXp-mstyle" id="MJXp-Span-41"><span class="MJXp-msub" id="MJXp-Span-42"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-43"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-44" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-10">B_2</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-11-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-45"><span class="MJXp-mstyle" id="MJXp-Span-46"><span class="MJXp-msub" id="MJXp-Span-47"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-48"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-49" style="vertical-align: -0.4em;">3</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-11">B_3</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-12-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-50"><span class="MJXp-mstyle" id="MJXp-Span-51"><span class="MJXp-msub" id="MJXp-Span-52"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-53"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-54" style="vertical-align: -0.4em;">4</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-12">B_4</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap="" rowspan="4">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-13-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-55"><span class="MJXp-mstyle" id="MJXp-Span-56"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-57">A</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-13">A</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-14-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-58"><span class="MJXp-mstyle" id="MJXp-Span-59"><span class="MJXp-msub" id="MJXp-Span-60"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-61"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-62" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-14">A_1</script>
</td>
<td class="MatrixTL"></td>
<td nowrap=""> 3 </td>
<td nowrap=""> 5 </td>
<td nowrap=""> 4 </td>
<td nowrap=""> 2 </td>
<td class="MatrixTR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-15-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-63"><span class="MJXp-mstyle" id="MJXp-Span-64"><span class="MJXp-msub" id="MJXp-Span-65"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-66"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-67" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-15">A_2</script>
</td>
<td class="MatrixL"></td>
<td nowrap=""> 5 </td>
<td nowrap=""> 6 </td>
<td nowrap=""> 2 </td>
<td nowrap=""> 4 </td>
<td class="MatrixR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-16-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-68"><span class="MJXp-mstyle" id="MJXp-Span-69"><span class="MJXp-msub" id="MJXp-Span-70"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-71"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-72" style="vertical-align: -0.4em;">3</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-16">A_3</script>
</td>
<td class="MatrixL"></td>
<td nowrap=""> 2 </td>
<td nowrap=""> 1 </td>
<td nowrap=""> 4 </td>
<td nowrap=""> 0 </td>
<td class="MatrixR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-17-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-73"><span class="MJXp-mstyle" id="MJXp-Span-74"><span class="MJXp-msub" id="MJXp-Span-75"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-76"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-77" style="vertical-align: -0.4em;">4</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-17">A_4</script>
</td>
<td class="MatrixBL"></td>
<td nowrap=""> 3 </td>
<td nowrap=""> 3 </td>
<td nowrap=""> 5 </td>
<td nowrap=""> 2 </td>
<td class="MatrixBR"></td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>7.3</b>
<a title="Algebraic method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=algebraic">Algebraic method</a>
</td>
</tr>
<tr align="left">
<td> 1. Find the solution of game using algebraic method for the following pay-off matrix <table border="0" cellspacing="0" cellpadding="5" style="border-collapse: collapse">
<tbody>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap="" colspan="2">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-18-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-78"><span class="MJXp-mstyle" id="MJXp-Span-79"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-80">B</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-18">B</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-19-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-81"><span class="MJXp-mstyle" id="MJXp-Span-82"><span class="MJXp-msub" id="MJXp-Span-83"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-84"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-85" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-19">B_1</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-20-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-86"><span class="MJXp-mstyle" id="MJXp-Span-87"><span class="MJXp-msub" id="MJXp-Span-88"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-89"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-90" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-20">B_2</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap="" rowspan="2">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-21-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-91"><span class="MJXp-mstyle" id="MJXp-Span-92"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-93">A</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-21">A</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-22-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-94"><span class="MJXp-mstyle" id="MJXp-Span-95"><span class="MJXp-msub" id="MJXp-Span-96"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-97"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-98" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-22">A_1</script>
</td>
<td class="MatrixTL"></td>
<td nowrap=""> 1 </td>
<td nowrap=""> 7 </td>
<td class="MatrixTR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-23-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-99"><span class="MJXp-mstyle" id="MJXp-Span-100"><span class="MJXp-msub" id="MJXp-Span-101"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-102"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-103" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-23">A_2</script>
</td>
<td class="MatrixBL"></td>
<td nowrap=""> 6 </td>
<td nowrap=""> 2 </td>
<td class="MatrixBR"></td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>7.4</b>
<a title="Calculus method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=calculus">Calculus method</a>
</td>
</tr>
<tr align="left">
<td> 1. Find the solution of game using calculus method for the following pay-off matrix <table border="0" cellspacing="0" cellpadding="5" style="border-collapse: collapse">
<tbody>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap="" colspan="2">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-24-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-104"><span class="MJXp-mstyle" id="MJXp-Span-105"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-106">B</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-24">B</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-25-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-107"><span class="MJXp-mstyle" id="MJXp-Span-108"><span class="MJXp-msub" id="MJXp-Span-109"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-110"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-111" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-25">B_1</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-26-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-112"><span class="MJXp-mstyle" id="MJXp-Span-113"><span class="MJXp-msub" id="MJXp-Span-114"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-115"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-116" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-26">B_2</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap="" rowspan="2">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-27-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-117"><span class="MJXp-mstyle" id="MJXp-Span-118"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-119">A</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-27">A</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-28-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-120"><span class="MJXp-mstyle" id="MJXp-Span-121"><span class="MJXp-msub" id="MJXp-Span-122"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-123"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-124" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-28">A_1</script>
</td>
<td class="MatrixTL"></td>
<td nowrap=""> 1 </td>
<td nowrap=""> 3 </td>
<td class="MatrixTR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-29-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-125"><span class="MJXp-mstyle" id="MJXp-Span-126"><span class="MJXp-msub" id="MJXp-Span-127"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-128"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-129" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-29">A_2</script>
</td>
<td class="MatrixBL"></td>
<td nowrap=""> 5 </td>
<td nowrap=""> 2 </td>
<td class="MatrixBR"></td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>7.5</b>
<a title="Arithmetic method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=arithmetic">Arithmetic method</a>
</td>
</tr>
<tr align="left">
<td> 1. Find the solution of game using arithmetic method for the following pay-off matrix <table border="0" cellspacing="0" cellpadding="5" style="border-collapse: collapse">
<tbody>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap="" colspan="3">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-30-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-130"><span class="MJXp-mstyle" id="MJXp-Span-131"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-132">B</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-30">B</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-31-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-133"><span class="MJXp-mstyle" id="MJXp-Span-134"><span class="MJXp-msub" id="MJXp-Span-135"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-136"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-137" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-31">B_1</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-32-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-138"><span class="MJXp-mstyle" id="MJXp-Span-139"><span class="MJXp-msub" id="MJXp-Span-140"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-141"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-142" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-32">B_2</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-33-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-143"><span class="MJXp-mstyle" id="MJXp-Span-144"><span class="MJXp-msub" id="MJXp-Span-145"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-146"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-147" style="vertical-align: -0.4em;">3</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-33">B_3</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap="" rowspan="3">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-34-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-148"><span class="MJXp-mstyle" id="MJXp-Span-149"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-150">A</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-34">A</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-35-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-151"><span class="MJXp-mstyle" id="MJXp-Span-152"><span class="MJXp-msub" id="MJXp-Span-153"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-154"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-155" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-35">A_1</script>
</td>
<td class="MatrixTL"></td>
<td nowrap=""> 10 </td>
<td nowrap=""> 5 </td>
<td nowrap=""> -2 </td>
<td class="MatrixTR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-36-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-156"><span class="MJXp-mstyle" id="MJXp-Span-157"><span class="MJXp-msub" id="MJXp-Span-158"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-159"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-160" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-36">A_2</script>
</td>
<td class="MatrixL"></td>
<td nowrap=""> 13 </td>
<td nowrap=""> 12 </td>
<td nowrap=""> 15 </td>
<td class="MatrixR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-37-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-161"><span class="MJXp-mstyle" id="MJXp-Span-162"><span class="MJXp-msub" id="MJXp-Span-163"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-164"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-165" style="vertical-align: -0.4em;">3</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-37">A_3</script>
</td>
<td class="MatrixBL"></td>
<td nowrap=""> 16 </td>
<td nowrap=""> 14 </td>
<td nowrap=""> 10 </td>
<td class="MatrixBR"></td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>7.6</b>
<a title="Matrix method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=matrix">Matrix method</a>
</td>
</tr>
<tr align="left">
<td> 1. Find the solution of game using matrix method for the following pay-off matrix <table border="0" cellspacing="0" cellpadding="5" style="border-collapse: collapse">
<tbody>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap="" colspan="3">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-38-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-166"><span class="MJXp-mstyle" id="MJXp-Span-167"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-168">B</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-38">B</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-39-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-169"><span class="MJXp-mstyle" id="MJXp-Span-170"><span class="MJXp-msub" id="MJXp-Span-171"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-172"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-173" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-39">B_1</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-40-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-174"><span class="MJXp-mstyle" id="MJXp-Span-175"><span class="MJXp-msub" id="MJXp-Span-176"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-177"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-178" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-40">B_2</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-41-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-179"><span class="MJXp-mstyle" id="MJXp-Span-180"><span class="MJXp-msub" id="MJXp-Span-181"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-182"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-183" style="vertical-align: -0.4em;">3</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-41">B_3</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap="" rowspan="3">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-42-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-184"><span class="MJXp-mstyle" id="MJXp-Span-185"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-186">A</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-42">A</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-43-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-187"><span class="MJXp-mstyle" id="MJXp-Span-188"><span class="MJXp-msub" id="MJXp-Span-189"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-190"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-191" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-43">A_1</script>
</td>
<td class="MatrixTL"></td>
<td nowrap=""> 1 </td>
<td nowrap=""> 7 </td>
<td nowrap=""> 2 </td>
<td class="MatrixTR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-44-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-192"><span class="MJXp-mstyle" id="MJXp-Span-193"><span class="MJXp-msub" id="MJXp-Span-194"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-195"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-196" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-44">A_2</script>
</td>
<td class="MatrixL"></td>
<td nowrap=""> 6 </td>
<td nowrap=""> 2 </td>
<td nowrap=""> 7 </td>
<td class="MatrixR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-45-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-197"><span class="MJXp-mstyle" id="MJXp-Span-198"><span class="MJXp-msub" id="MJXp-Span-199"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-200"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-201" style="vertical-align: -0.4em;">3</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-45">A_3</script>
</td>
<td class="MatrixBL"></td>
<td nowrap=""> 5 </td>
<td nowrap=""> 1 </td>
<td nowrap=""> 6 </td>
<td class="MatrixBR"></td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>7.7</b>
<a title="2Xn Games" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=2xn">2Xn Games</a>
</td>
</tr>
<tr align="left">
<td> 1. Find the solution of game using 2Xn Games method for the following pay-off matrix <table border="0" cellspacing="0" cellpadding="5" style="border-collapse: collapse">
<tbody>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap="" colspan="2">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-46-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-202"><span class="MJXp-mstyle" id="MJXp-Span-203"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-204">B</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-46">B</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-47-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-205"><span class="MJXp-mstyle" id="MJXp-Span-206"><span class="MJXp-msub" id="MJXp-Span-207"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-208"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-209" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-47">B_1</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-48-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-210"><span class="MJXp-mstyle" id="MJXp-Span-211"><span class="MJXp-msub" id="MJXp-Span-212"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-213"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-214" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-48">B_2</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap="" rowspan="3">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-49-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-215"><span class="MJXp-mstyle" id="MJXp-Span-216"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-217">A</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-49">A</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-50-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-218"><span class="MJXp-mstyle" id="MJXp-Span-219"><span class="MJXp-msub" id="MJXp-Span-220"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-221"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-222" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-50">A_1</script>
</td>
<td class="MatrixTL"></td>
<td nowrap=""> -3 </td>
<td nowrap=""> 4 </td>
<td class="MatrixTR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-51-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-223"><span class="MJXp-mstyle" id="MJXp-Span-224"><span class="MJXp-msub" id="MJXp-Span-225"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-226"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-227" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-51">A_2</script>
</td>
<td class="MatrixL"></td>
<td nowrap=""> -1 </td>
<td nowrap=""> 1 </td>
<td class="MatrixR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-52-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-228"><span class="MJXp-mstyle" id="MJXp-Span-229"><span class="MJXp-msub" id="MJXp-Span-230"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-231"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-232" style="vertical-align: -0.4em;">3</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-52">A_3</script>
</td>
<td class="MatrixBL"></td>
<td nowrap=""> 7 </td>
<td nowrap=""> -2 </td>
<td class="MatrixBR"></td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>7.8</b>
<a title="Graphical method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=graph">Graphical method</a>
</td>
</tr>
<tr align="left">
<td> 1. Find the solution of game using graphical method method for the following pay-off matrix <table border="0" cellspacing="0" cellpadding="5" style="border-collapse: collapse">
<tbody>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap="" colspan="2">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-53-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-233"><span class="MJXp-mstyle" id="MJXp-Span-234"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-235">B</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-53">B</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-54-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-236"><span class="MJXp-mstyle" id="MJXp-Span-237"><span class="MJXp-msub" id="MJXp-Span-238"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-239"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-240" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-54">B_1</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-55-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-241"><span class="MJXp-mstyle" id="MJXp-Span-242"><span class="MJXp-msub" id="MJXp-Span-243"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-244"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-245" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-55">B_2</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap="" rowspan="6">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-56-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-246"><span class="MJXp-mstyle" id="MJXp-Span-247"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-248">A</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-56">A</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-57-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-249"><span class="MJXp-mstyle" id="MJXp-Span-250"><span class="MJXp-msub" id="MJXp-Span-251"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-252"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-253" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-57">A_1</script>
</td>
<td class="MatrixTL"></td>
<td nowrap=""> 1 </td>
<td nowrap=""> -3 </td>
<td class="MatrixTR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-58-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-254"><span class="MJXp-mstyle" id="MJXp-Span-255"><span class="MJXp-msub" id="MJXp-Span-256"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-257"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-258" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-58">A_2</script>
</td>
<td class="MatrixL"></td>
<td nowrap=""> 3 </td>
<td nowrap=""> 5 </td>
<td class="MatrixR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-59-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-259"><span class="MJXp-mstyle" id="MJXp-Span-260"><span class="MJXp-msub" id="MJXp-Span-261"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-262"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-263" style="vertical-align: -0.4em;">3</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-59">A_3</script>
</td>
<td class="MatrixL"></td>
<td nowrap=""> -1 </td>
<td nowrap=""> 6 </td>
<td class="MatrixR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-60-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-264"><span class="MJXp-mstyle" id="MJXp-Span-265"><span class="MJXp-msub" id="MJXp-Span-266"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-267"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-268" style="vertical-align: -0.4em;">4</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-60">A_4</script>
</td>
<td class="MatrixL"></td>
<td nowrap=""> 4 </td>
<td nowrap=""> 1 </td>
<td class="MatrixR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-61-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-269"><span class="MJXp-mstyle" id="MJXp-Span-270"><span class="MJXp-msub" id="MJXp-Span-271"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-272"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-273" style="vertical-align: -0.4em;">5</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-61">A_5</script>
</td>
<td class="MatrixL"></td>
<td nowrap=""> 2 </td>
<td nowrap=""> 2 </td>
<td class="MatrixR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-62-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-274"><span class="MJXp-mstyle" id="MJXp-Span-275"><span class="MJXp-msub" id="MJXp-Span-276"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-277"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-278" style="vertical-align: -0.4em;">6</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-62">A_6</script>
</td>
<td class="MatrixBL"></td>
<td nowrap=""> -5 </td>
<td nowrap=""> 0 </td>
<td class="MatrixBR"></td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>7.9</b>
<a title="LPP method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=lpp">LPP method</a>
</td>
</tr>
<tr align="left">
<td> 1. Find the solution of game using linear programming method for the following pay-off matrix <table border="0" cellspacing="0" cellpadding="5" style="border-collapse: collapse">
<tbody>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap="" colspan="3">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-63-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-279"><span class="MJXp-mstyle" id="MJXp-Span-280"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-281">B</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-63">B</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""></td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-64-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-282"><span class="MJXp-mstyle" id="MJXp-Span-283"><span class="MJXp-msub" id="MJXp-Span-284"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-285"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-286" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-64">B_1</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-65-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-287"><span class="MJXp-mstyle" id="MJXp-Span-288"><span class="MJXp-msub" id="MJXp-Span-289"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-290"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-291" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-65">B_2</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-66-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-292"><span class="MJXp-mstyle" id="MJXp-Span-293"><span class="MJXp-msub" id="MJXp-Span-294"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-295"
style="margin-right: 0.05em;">B</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-296" style="vertical-align: -0.4em;">3</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-66">B_3</script>
</td>
<td nowrap=""></td>
<td nowrap=""></td>
</tr>
<tr nowrap="" align="center">
<td nowrap="" rowspan="3">Player <span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-67-Frame" tabindex="0"><span
class="MJXp-math" id="MJXp-Span-297"><span class="MJXp-mstyle" id="MJXp-Span-298"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-299">A</span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-67">A</script>
</td>
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-68-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-300"><span class="MJXp-mstyle" id="MJXp-Span-301"><span class="MJXp-msub" id="MJXp-Span-302"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-303"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-304" style="vertical-align: -0.4em;">1</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-68">A_1</script>
</td>
<td class="MatrixTL"></td>
<td nowrap=""> 3 </td>
<td nowrap=""> -4 </td>
<td nowrap=""> 2 </td>
<td class="MatrixTR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-69-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-305"><span class="MJXp-mstyle" id="MJXp-Span-306"><span class="MJXp-msub" id="MJXp-Span-307"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-308"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-309" style="vertical-align: -0.4em;">2</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-69">A_2</script>
</td>
<td class="MatrixL"></td>
<td nowrap=""> 1 </td>
<td nowrap=""> -7 </td>
<td nowrap=""> -3 </td>
<td class="MatrixR"></td>
</tr>
<tr nowrap="" align="center">
<td nowrap=""><span class="MathJax_Preview" style="display: none;"></span><span class="MathJax_PHTML" id="MathJax-Element-70-Frame" tabindex="0"><span class="MJXp-math"
id="MJXp-Span-310"><span class="MJXp-mstyle" id="MJXp-Span-311"><span class="MJXp-msub" id="MJXp-Span-312"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-313"
style="margin-right: 0.05em;">A</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-314" style="vertical-align: -0.4em;">3</span></span></span></span></span>
<script type="math/asciimath" id="MathJax-Element-70">A_3</script>
</td>
<td class="MatrixBL"></td>
<td nowrap=""> -2 </td>
<td nowrap=""> 4 </td>
<td nowrap=""> 7 </td>
<td class="MatrixBR"></td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
<td align="center" valign="top">
<table border="0" width="100%" cellspacing="2" cellpadding="1">
<tbody>
<tr class="solutiontext">
<td align="center">
<b>7.10</b>
<a title="Bimatrix method" href="//cbom.atozmath.com/CBOM/GameTheory.aspx?q=bimatrix">Bimatrix method</a>
</td>
</tr>
<tr align="left">
<td>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
</tbody>
</table>
</td>
</tr>
<tr>
<td align="center" valign="top" colspan="2"> </td>
</tr>
<tr>
<td align="center" valign="top">
</td>
<td align="center" valign="top">
</td>
</tr>
</tbody>
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Learn more Support us (New) All problem can be solved using search box I want to sell my website www.AtoZmath.com with complete code Powered by Google Übersetzer Home What's new College Algebra Games Feedback About us Algebra Matrix & Vector Numerical Methods Statistical Methods Operation Research Word Problems Calculus Geometry Pre-Algebra AtoZmath1 00:00/00:00 AtoZmath1 AtoZmath2 AtoZmath3 -------------------------------------------------------------------------------- Home > Operation Research calculators -------------------------------------------------------------------------------- AtoZmath.com - Homework help (with all solution steps) Educational Level Secondary school, High school and College Program Purpose Provide step by step solutions of your problems using online calculators (online solvers) Problem Source Your textbook, etc Operation Research Calculators (examples) 1. Assignment problem 1.1 Assignment problem (Using Hungarian method-2) 1.2 Assignment problem (Using Hungarian method-1) 2.1 Travelling salesman problem using hungarian method 2.2 Travelling salesman problem using branch and bound (penalty) method 2.3 Travelling salesman problem using branch and bound method 2.4 Travelling salesman problem using nearest neighbor method 2.5 Travelling salesman problem using diagonal completion method 3. Crew assignment problem -------------------------------------------------------------------------------- 2. Simplex method (Solve linear programming problem using) 0. Formulate linear programming model examples 1. Graphical method 2. Simplex method (BigM method) 3. Two-Phase method 4. Primal to dual conversion 5. Dual Simplex method 6. Integer Simplex method (Gomory's cutting plane method) 7. Branch and Bound method 8. 0-1 Integer programming problem 9. Revised Simplex method -------------------------------------------------------------------------------- -------------------------------------------------------------------------------- 3. Transportation Problem using 1. North-West corner method 2. Least cost method 3. Vogel's approximation method 4. Row minima method 5. Column minima method 6. Russell's approximation method 7. Heuristic method-1 8. Heuristic method-2 9. Optimal solution using MODI method 10. Optimal solution using stepping stone method -------------------------------------------------------------------------------- 4. PERT and CPM 1. Network diagram 1. Activity, Predecessors 2. Activity i-j 3. Activity i-j, Name of Activity 2. Critical path, Total float, Free float, Independent float 1. Activity, Predecessors, Duration 2. Activity i-j, Duration 3. Activity i-j, Name of Activity, Duration 3. Project scheduling with uncertain activity times (Optimistic, Most likely, Pessimistic) 1. Activity, Predecessors, to, tm, tp 2. Activity i-j, to, tm, tp 3. Activity i-j, Name of Activity, to, tm, tp 4. Project crashing to solve Time-Cost Trade-Off with fixed Indirect cost 1. Activity, Predecessors, Normal Time & Cost, Crash Time & Cost and Indirect Cost 2. Activity i-j, Normal Time & Cost, Crash Time & Cost and Indirect Cost 3. Activity i-j, Name of Activity, Normal Time & Cost, Crash Time & Cost and Indirect Cost 5. Project crashing to solve Time-Cost Trade-Off with varying Indirect cost 1. Activity, Predecessors, Normal Time & Cost, Crash Time & Cost and varying Indirect Cost 2. Activity i-j, Normal Time & Cost, Crash Time & Cost and varying Indirect Cost 3. Activity i-j, Name of Activity, Normal Time & Cost, Crash Time & Cost and varying Indirect Cost -------------------------------------------------------------------------------- 5. Sequencing Problems 1. Processing n Jobs Through 2 Machines Problem 2. Processing n Jobs Through 3 Machines Problem 3. Processing n Jobs Through m Machines Problem 4. Processing 2 Jobs Through m Machines Problem -------------------------------------------------------------------------------- -------------------------------------------------------------------------------- 6. Replacement and Maintenance Models 1. Model-1 : Replacement policy for items whose running cost increases with time and value of money remains constant during a period 1.1 Model-1.1 1.2 Model-1.2 1.3 Model-1.3 2. Model-2 : Replacement policy for items whose running cost increases with time but value of money changes constant rate during a period 3. Model-3 : Group replacement policy -------------------------------------------------------------------------------- 7. Game Theory 1. Saddle Point 2. Dominance method 3. Oddment method 4. Algebraic method 5. Calculus method 6. Arithmetic method 7. Matrix method 8. 2Xn Games 9. Graphical method 10. LPP method 11. Bimatrix method -------------------------------------------------------------------------------- 8. Data envelopment analysis (DEA method) -------------------------------------------------------------------------------- Operation Research with example 1. Assignment Problem 1.1 Balanced Assignment Problem (Using Hungarian method) 1. A department has five employess with five jobs to be permormed. The time (in hours) each men will take to perform each job is given in the effectiveness matrix. Employees I II III IV V Jobs A 10 5 113 15 16 B 3 9 18 13 6 C 10 7 2 2 2 D 7 11 9 7 12 E 7 9 10 4 12 How should the jobs be allocated, one per employee, so as to minimize the total man-hours? 1.2 Unbalanced Assignment Problem (Using Hungarian method) 2. In the modification of a plant layout of a factory four new machines M1, M2, M3 and M4 are to be installed in a machine shop. There are five vacant places A, B, C, D and E available. Because of limited space, machine M2 cannot be placed at C and M3 cannot be placed at A. The cost of locating a machine at a place (in hundred rupess) is as follows. Location A B C D E Machine M1 9 11 15 10 11 M2 12 9 -- 10 9 M3 -- 11 14 11 7 M4 14 8 12 7 8 Find the optimal assignment schedule. 2. Travelling salesman problem 2.1 using hungarian method 2.2 using branch and bound (penalty) method 2.3 using branch and bound method 2.4 Travelling salesman problem using nearest neighbor method 2.5 Travelling salesman problem using diagonal completion method 1. A travelling salesman has to visit five cities. He wishes to start from a particular city, visit each city only once and then return to his starting point. The travelling cost of each city from a particular city is given below. To city A B C D E From city A x 2 5 7 1 B 6 x 3 8 2 C 8 7 x 4 7 D 12 4 6 x 5 E 1 3 2 8 x How should the jobs be allocated, one per employee, so as to minimize the total man-hours? 3 Crew assignment problem 1. Best-ride airlines that operates seven days a week has the following time-table. Delhi - Mumbai Mumbai - Delhi Flight No Departure Arrival 1 7.00 8.00 2 8.00 9.00 3 13.00 14.00 4 18.00 19.00 Flight No Departure Arrival 101 8.00 9.00 102 9.00 10.00 103 12.00 13.00 104 17.00 18.00 Crews must have a minimum layover of 5 hours between flights. Obtain the pairing of flights that minimizes layover time away from home. For any given pairing, the crew will be based at the city that results in the smaller layover. For each pair also mention the city where crew should be based. 2. Simplex method Solve the following LP problem by using 1. Simplex method (BigM method) 2. Two-Phase method 3. Graphical method 4. Primal to dual conversion 5. Dual Simplex method 6. Integer Simplex method (Gomory's cutting plane method) 7. Branch and Bound method 8. 0-1 Integer programming problem 9. Revised Simplex method 2.1 Simplex method 1. Use the simplex method to solve the following LP problem. Maximize Z = 3x1 + 5x2 + 4x3 subject to the constraints 2x1 + 3x2 ≤ 8 2x2 + 5x3 ≤ 10 3x1 + 2x2 + 4x3 ≤ 15 and x1, x2, x3 ≥ 0 2. Use the simplex method to solve the following LP problem. Maximize Z = 4x1 + 3x2 subject to the constraints 2x1 + x2 ≤ 1000 x1 + x2 ≤ 800 x1 ≤ 400 x2 ≤ 700 and x1, x2 ≥ 0 2.2 BigM method 1. Use the penalty (Big - M) method to solve the following LP problem. Minimize Z = 5x1 + 3x2 subject to the constraints 2x1 + 4x2 ≤ 12 2x1 + 2x2 = 10 5x1 + 2x2 ≥ 10 and x1, x2 ≥ 0 2. Use the penalty (Big - M) method to solve the following LP problem. Minimize Z = x1 + 2x2 + 3x3 - x4 subject to the constraints x1 + 2x2 + 3x3 = 15 2x1 + x2 + 5x3 = 20 x1 + 2x2 + x3 + x4 = 10 and x1, x2, x3, x4 ≥ 0 2.3 Two-Phase method 1. Solve the following LP problem by using the Two-Phase method. Minimize Z = x1 + x2 subject to the constraints 2x1 + 4x2 ≥ 4 x1 + 7x2 ≥ 7 and x1, x2 ≥ 0 2. Solve the following LP problem by using the Two-Phase method. Minimize Z = x1 - 2x2 - 3x3 subject to the constraints -2x1 + 3x2 + 3x3 = 2 2x1 + 3x2 + 4x3 = 1 and x1, x2, x3 ≥ 0 2.4 Dual Simplex method 1. Solve the following LP problem by using the Two-Phase method. Minimize Z = x1 + x2 subject to the constraints 2x1 + 4x2 ≥ 4 x1 + 7x2 ≥ 7 and x1, x2 ≥ 0 2. Solve the following LP problem by using the Two-Phase method. Minimize Z = x1 - 2x2 - 3x3 subject to the constraints -2x1 + 3x2 + 3x3 = 2 2x1 + 3x2 + 4x3 = 1 and x1, x2, x3 ≥ 0 2.5 Gomorys Integer Cutting method 1. Solve the following integer programming problem using Gomory's cutting plane algorithm. Maximize Z = x1 + x2 subject to the constraints 3x1 + 2x2 ≤ 5 x2 ≤ 2 and x1, x2 ≥ 0 and are integers. 2. Solve the following integer programming problem using Gomory's cutting plane algorithm. Maximize Z = 2x1 + 20x2 - 10x3 subject to the constraints 2x1 + 20x2 + 4x3 ≤ 15 6x1 + 20x2 + 4x3 ≤ 20 and x1, x2, x3 ≥ 0 and are integers. 2.6 Graphical method 1. Use graphical method to solve following LP problem. Maximize Z = x1 + x2 subject to the constraints 3x1 + 2x2 ≤ 5 x2 ≤ 2 and x1, x2 ≥ 0 2. Use graphical method to solve following LP problem. Maximize Z = 2x1 + x2 subject to the constraints x1 + 2x2 ≤ 10 x1 + x2 ≤ 6 x1 - x2 ≤ 2 x1 - 2x2 ≤ 1 and x1, x2 ≥ 0 2.7 Primal to dual conversion 1. Write the dual to the following LP problem. Maximize Z = x1 - x2 + 3x3 subject to the constraints x1 + x2 + x3 ≤ 10 2x1 - x2 - x3 ≤ 2 2x1 - 2x2 - 3x3 ≤ 6 and x1, x2, x3 ≥ 0 2. Write the dual to the following LP problem. Minimize Z = 3x1 - 2x2 + 4x3 subject to the constraints 3x1 + 5x2 + 4x3 ≥ 7 6x1 + x2 + 3x3 ≥ 4 7x1 - 2x2 - x3 ≤ 10 x1 - 2x2 + 5x3 ≥ 3 4x1 + 7x2 - 2x3 ≥ 2 and x1, x2, x3 ≥ 0 2.8 Branch and Bound method 1. Solve the following LP problem by using Branch and Bound method Max Z = 7x1 + 9x2 subject to -x1 + 3x2 ≤ 6 7x1 + x2 ≤ 35 x2 ≤ 7 and x1,x2 ≥ 0 2. Solve the following LP problem by using Branch and Bound method Max Z = 3x1 + 5x2 subject to 2x1 + 4x2 ≤ 25 x1 ≤ 8 2x2 ≤ 10 and x1,x2 ≥ 0 2.9 0-1 Integer programming problem 1. Solve LP using zero-one Integer programming problem method Max Z = 300x1 + 90x2 + 400x3 + 150x4 subject to 35000x1 + 10000x2 + 25000x3 + 90000x4 ≤ 120000 4x1 + 2x2 + 7x3 + 3x4 ≤ 12 x1 + x2 ≤ 1 and x1,x2,x3,x4 ≥ 0 2. Solve LP using 0-1 Integer programming problem method MAX Z = 650x1 + 700x2 + 225x3 + 250x4 subject to 700x1 + 850x2 + 300x3 + 350x4 ≤ 1200 550x1 + 550x2 + 150x3 + 200x4 ≤ 700 400x1 + 350x2 + 100x3 ≤ 400 x1 + x2 ≥ 1 -x3 + x4 ≤ 1 and x1,x2,x3,x4 ≥ 0 2.9 Revised Simplex method 1. Solve the following LP problem by using Revised Simplex method MAX Z = 3x1 + 5x2 subject to x1 ≤ 4 x2 ≤ 6 3x1 + 2x2 ≤ 18 and x1,x2 ≥ 0 2. Solve the following LP problem by using Revised Simplex method MAX Z = 2x1 + x2 subject to 3x1 + 4x2 ≤ 6 6x1 + x2 ≤ 3 and x1,x2 ≥ 0 3. Transportation Problem using 1. North-West corner method 2. Least cost method 3. Vogel's approximation method 4. Row minima method 5. Column minima method 6. Russell's approximation method 7. Heuristic method-1 8. Heuristic method-1 9. optimal solution by MODI method 10. Optimal solution using stepping stone method 1. A Company has 3 production facilities S1, S2 and S3 with production capacity of 7, 9 and 18 units (in 100's) per week of a product, respectively. These units are tobe shipped to 4 warehouses D1, D2, D3 and D4 with requirement of 5,6,7 and 14 units (in 100's) per week, respectively. The transportation costs (in rupees) per unit between factories to warehouses are given in the table below. D1 D2 D3 D4 Capacity S1 19 30 50 10 7 S2 70 30 40 60 9 S3 40 8 70 20 18 Demand 5 8 7 14 34 Find initial basic feasible solution for given problem by using (a) North-West corner method (b) Least cost method (c) Vogel's approximation method (d) obtain an optimal solution by MODI method if the object is to minimize the total transportation cost. 2. Find an initial basic feasible solution for given transportation problem by using (a) North-West corner method (b) Least cost method (c) Vogel's approximation method D1 D2 D3 D4 Supply S1 11 13 17 14 250 S2 16 18 14 10 300 S3 21 24 13 10 400 Demand 200 225 275 250 3. A company has factories at F1, F2 and F3 which supply to warehouses at W1, W2 and W3. Weekly factory capacities are 200, 160 and 90 units, respectively. Weekly warehouse requiremnet are 180, 120 and 150 units, respectively. Unit shipping costs (in rupess) are as follows: W1 W2 W3 Supply F1 16 20 12 200 F2 14 8 18 160 F3 26 24 16 90 Demand 180 120 150 450 Determine the optimal distribution for this company to minimize total shipping cost. 4. Find an initial basic feasible solution for given transportation problem by using (a) North-West corner method (b) Least cost method (c) Vogel's approximation method P Q R S Supply A 6 3 5 4 22 B 5 9 2 7 15 C 5 7 8 6 8 Demand 7 12 17 9 45 4. PERT and CPM 1. An assembly is to be made from two parts X and Y. Both parts must be turned on a lathe Y must be polished where as X need not be polished. The sequence of acitivities, together with their predecessors, is given below Activity Description Predecessor Activity A Open work order - B Get material for X A C Get material for Y A D Turn X on lathe B E Turn Y on lathe B,C F Polish Y E G Assemble X and Y D,F H Pack G Draw a network diagram of activities for the project. 2. An established company has decided to add a new product to its line. It will buy the product from a manufacturing concern, package it, and sell it to a number of distributors that have been selected on a geographical basis. Market research has already indicated the volume expected and the size of sales force required. The steps shown in the following table are to be planned. Activity Description Predecessor Activity Duration (days) A Organize sales office - 6 B Hire salesman A 4 C Train salesman B 7 D Select advertising agency A 2 E Plan advertising campaign D 4 F Conduct advertising campaign E 10 G Design package - 2 H Setup packaging campaign G 10 I Package initial stocks J,H 6 J Order stock from manufacturer - 13 K Select distributors A 9 L Sell to distributors C,K 3 M Ship stocks to distributors I,L 5 (a) Draw an arrow diagram for the project. (b) Indicate the criticla path. (c) For each non-critical activity, find the total and free float. 5. Sequencing Problems 1. There are seven jobs, each of which has to go through the machines A and B in the order AB. Processing times in hours are as follows. Job 1 2 3 4 5 6 7 Machine A 3 12 15 6 10 11 9 Machine B 8 10 10 6 12 1 3 Decide a sequence of these jobs that will minimize the total elapsed time T. Also find T and idle time for machines A and B. 2. Find the sequence that minimizes the total time required in performing the following job on three machines in the order ABC. Processing times (in hours) are given in the following table. Job 1 2 3 4 5 Machine A 8 10 6 7 11 Machine B 5 6 2 3 4 Machine C 4 9 8 6 5 6. Replacement and Maintenance Models Model-1.1 1. A firm is considering the replacement of a machine, whose cost price is Rs 12,200 and its scrap value is Rs 200. From experience the running (maintenance and operating) costs are found to be as follows: Year12345678Running Cost2005008001,2001,8002,5003,2004,000 When should the machine be replaced? Model-1.2 1. The data collected in running a machine, the cost of which is Rs 60,000 are given below: Year12345Resale Value42,00030,00020,40014,4009,650Cost of spares4,0004,2704,8805,7006,800Cost of labour14,00016,00018,00021,00025,000 Determine the optimum period for replacement of the machine. Model-1.3 1. Machine A costs Rs 45,000 and its operating costs are estimated to be Rs 1,000 for the first year increasing by Rs 10,000 per year in the second and subsequent years. Machine B costs Rs 50,000 and operating costs are Rs 2,000 for the first year, increasing by Rs 4,000 in the second and subsequent years. If at present we have a machine of type A, should we replace it with B? if so when? Assume that both machines have no resale value and their future costs are not discounted. Model-2 Replacement policy for items whose running cost increases with time but value of money changes constant rate during a period 1. An engineering company is offered a material handling equipment A. It is priced at Rs 60,000 includeing cost of installation. The costs for operation and maintenance are estimated to be Rs 10,000 for each of the first five years, increasing every year by Rs 3,000 in the sixth and subsequent years. The company expects a return of 10 percent on all its investment. What is the optimal replacement period? Year1234567Running Cost10,00010,00010,00010,00010,00013,00016,000 2. A company is buying mini computers. It costs Rs 5 lakh, and its running and maintenance costs are Rs 60,000 for each of the first five years, increasing by Rs 20,000 per year in the sixth and subsequent years. If the money is worth 10 percent per year, What is the optimal replacement period? Model-3 Group replacement policy 1. A computer contains 10,000 resistors. When any resistor fails, it is replaced. The cost of replacing a resistor individually is Rs 1 only. If all the resistors are replaced at the same time, the cost per resistor would be reduced to 35 paise. The percentage of surviving resistors say S(t) at the end of month t and the probability of failure P(t) during the month t are as follows: t0123456P(t)00.030.070.200.400.150.15 What is the optimal replacement plan? 2. The following mortality rates have been observed for a certain type of fuse: t012345P(t)00.050.100.200.400.25 There are 1,000 fuses in use and it costs Rs 5 to replace an individual fuse. If all fuses were replaced simultaneously it would cost Rs 1.25 per fuse. It is proposed to replace all fuses at fixed intervals of time, whether or not they have burnt out, and to contiune replacing burnt out fuses as they fail. At what time intervals should the group replacement be made? Also prove that this optimal policy is superior to the straight forward policy of replacing each fuse only when it fails. 7. Game Theory 1. Saddle Point 2. Dominance method 3. Oddment method 4. Algebraic method 5. Calculus method 6. Arithmetic method 7. Matrix method 8. 2Xn Games 9. Graphical method 10. LPP method 11. Bimatrix method 7.1 Saddle Point 1. For the game with payoff matrix Player BB1B2B3Player AA1 -1 2 -2 A2 6 4 -6 determine the best strategies for players A and B. Also determine the value of game. Is this game saddle point? 7.2 Dominance method 1. Dominance Example Player BB1B2B3B4Player AA1 3 5 4 2 A2 5 6 2 4 A3 2 1 4 0 A4 3 3 5 2 7.3 Algebraic method 1. Find the solution of game using algebraic method for the following pay-off matrix Player BB1B2Player AA1 1 7 A2 6 2 7.4 Calculus method 1. Find the solution of game using calculus method for the following pay-off matrix Player BB1B2Player AA1 1 3 A2 5 2 7.5 Arithmetic method 1. Find the solution of game using arithmetic method for the following pay-off matrix Player BB1B2B3Player AA1 10 5 -2 A2 13 12 15 A3 16 14 10 7.6 Matrix method 1. Find the solution of game using matrix method for the following pay-off matrix Player BB1B2B3Player AA1 1 7 2 A2 6 2 7 A3 5 1 6 7.7 2Xn Games 1. Find the solution of game using 2Xn Games method for the following pay-off matrix Player BB1B2Player AA1 -3 4 A2 -1 1 A3 7 -2 7.8 Graphical method 1. Find the solution of game using graphical method method for the following pay-off matrix Player BB1B2Player AA1 1 -3 A2 3 5 A3 -1 6 A4 4 1 A5 2 2 A6 -5 0 7.9 LPP method 1. Find the solution of game using linear programming method for the following pay-off matrix Player BB1B2B3Player AA1 3 -4 2 A2 1 -7 -3 A3 -2 4 7 7.10 Bimatrix method -------------------------------------------------------------------------------- -------------------------------------------------------------------------------- Share this solution or page with your friends. Home What's new College Algebra Games Feedback About us Copyright © 2023. All rights reserved. Terms, Privacy . Originaltext Diese Übersetzung bewerten Mit deinem Feedback können wir Google Übersetzer weiter verbessern 00:00/00:00