8167 /www/nrich/html/content/id/8167/ 5 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"><br></br> <p><span class="editorial">This is the curriculum mapping page for A-level decision mathematics topics.</span></p> <p><br></br>  </p> <table style="" border="1"> <tbody> <tr height="31"> <td bgcolor="#66FFFF" height="31" style="">AREA</td> <td>TOPICS COVERED</td> </tr> <tr height="31"> <td bgcolor="#66FFFF" height="31" style="">Decision D1</td> <td>Algorithms; algorithms on graphs; route inspection problem; critical path analysis; linear programming; matchings.</td> </tr> <tr height="31"> <td bgcolor="#66FFFF" height="31" style="">Decision D2</td> <td>Transportation problems; allocation (assignment) problems; the travelling salesman; game theory; further linear programming, dynamic programming; flows in networks.</td> </tr> </tbody> </table> <br></br> <br></br> <table style="" border="1"> <colgroup> <col width="400"></col> <col width="164"></col></colgroup> <tbody> <tr height="31"> <td bgcolor="#6699FF" colspan="2" height="31" width="400" style=""> <div>Decision D1</div> <div>Algorithms; algorithms on graphs; route inspection problem; critical path analysis; linear programming; matchings.</div> </td> </tr> <tr height="17"> <td bgcolor="#66FFFF" colspan="2" height="17" width="400" style="">ALGORITHMS</td> </tr> <tr height="34"> <td height="34" width="400" style=""> <p>The general ideas of algorithms and the<br></br> implementation of an algorithm given by a flow chart<br></br> or text. Bin packing, bubble sort, quick sort, binary search.</p> </td> <td width="164" style=""> <div><a href="/5956">Zeller&#39;s Birthday</a></div> <div><a href="http://nrich.maths.org/5918">Flow Chart</a></div> <div><a href="http://nrich.maths.org/5928">Procedure Solver</a></div> </td> </tr> <tr height="17"> <td bgcolor="#66FFFF" colspan="2" height="17" width="400" style="">ALGORITHMS ON GRAPHS</td> </tr> <tr height="17"> <td height="17" width="400" style="">The minimum spanning tree (minimum connector) problem. Prim’s and Kruskal’s (greedy) algorithm. Dijkstra’s algorithm for finding the shortest path.</td> <td width="164" style=""> </td> </tr> <tr height="17"> <td bgcolor="#66FFFF" colspan="2" height="17" width="400" style="">ROUTE INSPECTION PROBLEM</td> </tr> <tr height="17"> <td height="17" width="400" style="">Algorithm for finding the shortest route around a network, travelling along every edge at least once and ending at the start vertex.</td> <td width="164" style=""> </td> </tr> <tr height="17"> <td bgcolor="#66FFFF" colspan="2" height="17" width="400" style="">CRITICAL PATH ANALYSIS</td> </tr> <tr height="17"> <td height="17" width="400" style="">Modelling of a project by an activity network, from a<br></br> precedence table. Completion of the precedence table for a given activity network.<br></br> Algorithm for finding the critical path. Earliest and latest event times. Earliest and latest start and finish times for activities.<br></br> Total float. Gantt (cascade) charts. Scheduling.</td> <td width="164" style=""> </td> </tr> <tr height="17"> <td bgcolor="#66FFFF" colspan="2" height="17" width="400" style="">LINEAR PROGRAMMING</td> </tr> <tr height="17"> <td height="17" width="400" style="">Formulation of problems as linear programs.<br></br> Graphical solution of two variable problems using<br></br> ruler and vertex methods<br></br> Consideration of problems where solutions must<br></br> have integer values.</td> <td width="164" style=""> </td> </tr> <tr height="17"> <td bgcolor="#66FFFF" colspan="2" height="17" width="400" style="">MATCHINGS</td> </tr> <tr height="17"> <td height="17" width="400" style="">Use of bipartite graphs for modelling matchings.<br></br> Complete matchings and maximal matchings.<br></br> Algorithm for obtaining a maximum matching.</td> <td> </td> </tr> </tbody> </table> <p> </p> <p> </p> <table style="" border="1"> <colgroup> <col width="400"></col> <col width="164"></col></colgroup> <tbody> <tr height="31"> <td bgcolor="#6699FF" colspan="2" height="31" width="400" style=""> <div>Decision D2</div> <div>Transportation problems; allocation (assignment) problems; the travelling salesman; game theory; further linear programming, dynamic programming; flows in networks.</div> <div> </div> </td> </tr> <tr height="17"> <td bgcolor="#66FFFF" colspan="2" height="17" width="400" style="">TRANSPORTATION PROBLEMS</td> </tr> <tr height="34"> <td height="34" width="400" style="">The north-west corner method for finding an initial<br></br> basic feasible solution.Use of the stepping-stone method for obtaining an<br></br> improved solution. Improvement indices.Formulation as a linear programming problem.</td> <td width="164" style=""> <div> </div> </td> </tr> <tr height="17"> <td bgcolor="#66FFFF" colspan="2" height="17" width="400" style="">ALLOCATION ( ASSIGNMENT) PROBLEMS</td> </tr> <tr height="17"> <td height="17" width="400" style="">Cost matrix reduction.Use of the Hungarian algorithm to find least cost<br></br> allocation.Modification of method to deal with a maximum<br></br> profit allocation.Formulation as a linear programming problem.</td> <td width="164" style=""> </td> </tr> <tr height="17"> <td bgcolor="#66FFFF" colspan="2" height="17" width="400" style="">TRAVELLING SALESMAN</td> </tr> <tr height="17"> <td height="17" width="400" style="">The practical and classical problems. The classical<br></br> problem for complete graphs satisfying the triangle<br></br> inequality.Determination of upper and lower bounds using<br></br> minimum spanning tree methods.The nearest neighbour algorithm.</td> <td width="164" style=""> </td> </tr> <tr height="17"> <td bgcolor="#66FFFF" colspan="2" height="17" width="400" style="">GAME THEORY</td> </tr> <tr height="17"> <td height="17" width="400" style="">Two person zero-sum games and the pay-off matrix.<br></br> Identification of play safe strategies and stable<br></br> solutions (saddle points).Reduction of pay-off matrices using dominance<br></br> arguments. Optimal mixed strategies for a game<br></br> with no stable solution.Conversion of 3 × 2 and 3 × 3 games to linear<br></br> programming problems.</td> <td width="164" style=""> </td> </tr> <tr height="17"> <td bgcolor="#66FFFF" colspan="2" height="17" width="400" style="">FURTHER LINEAR PROGRAMMING</td> </tr> <tr height="17"> <td height="17" width="400" style=""> <p>Formulation of problems as linear programs.</p> <p>The Simplex algorithm and tableau for maximising<br></br> problems.The use and meaning of slack variables.</p> </td> <td width="164" style=""> </td> </tr> <tr height="17"> <td bgcolor="#66FFFF" colspan="2" height="17" width="400" style="">DYNAMIC PROGRAMMING</td> </tr> <tr height="17"> <td height="17" width="400" style="">Principles of dynamic programming. Bellman’s<br></br> principle of optimality.Stage variables and State variables. Use of tabulation to solve maximum, minimum, minimax or maximin problems.</td> <td> </td> </tr> <tr height="17"> <td bgcolor="#66FFFF" colspan="2" height="17" width="400" style="">FLOWS IN NETWORKS</td> </tr> <tr height="17"> <td height="17" width="400" style="">Algorithm for finding a maximum flow. Cuts and their capacity. Use of the max flow — min cut theorem to verify that a flow is a maximum flow. Multiple sources and sinks.</td> <td> </td> </tr> </tbody> </table> <p> </p> <p>NOTE: These NRICH problems often draw together strands from decision mathematics as a whole, and any one activity may have several possible lesson foci. For example, some problems will work well either as introduction to a topic to raise the underlying issues or as an end of module consolidation in which students&#39; problem solving and understanding is drawn together. Others are suitable for classroom discussion and others for individual problem solving.<br></br> <br></br> <br></br> <span style="font-style: italic;">The topics are grouped according to a summary of the Edexcel schemes, but should be generally of use.</span> <span style="font-style: italic;">Please note that this material is under development. New problems will arrive on a regular basis and it is intended that detailed teachers&#39; notes will eventually accompany all of the problems. If you have any comments please do get in touch</span>.<br></br>  </p></mdoxml> 0 3 0 0 0 0 1 This is the curriculum mapping page for decision modules