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11-1CHAPTER 11: DYNAMIC PROGRAMMING11.2-1.(a) The nodes of the network can be divided into "layers" such that the nodes in the th8layer are accessible from the origin only through the nodes in the st layer. TheseÐ8 "Ñlayers define the stages of the problem, which can be labeled as . The nodes8 œ "ß #ß $ß %constitute the states.Let denote the set of the nodes in the th layer of the network, i.e., , W8Wœ ÖS× Wœ8"#ÖEß Fß G×Wœ ÖHß I×Wœ ÖX×B, and . The decision variable is the immediate$%8destination at stage . Then the problem can be formulated as follows:80 Ð=Ñ œÒ- 0ÐB ÑÓ ´0 Ð=ß B Ñ= − W8 œ "ß #ß $‡‡88"B −WB −W=B8888for and minmin88"88"80 ÐXÑ œ !‡%(b) The shortest path is .S F H X(c) Number of stages: 36=0 Ð=ÑBHXI(X$‡‡$$=0 Ð=ß HÑ0 Ð=ß IÑ0 Ð=ÑBE""""HF"$"&"$HG"$"$I###‡‡##=0 Ð=ß EÑ0 Ð=ß FÑ0 Ð=ß GÑ0 Ð=ÑBS#!"*#!"*F""""‡‡""Optimal Solution: , and .Bœ F Bœ HBœ H‡‡‡"#$
11-2(d) Shortest-Path Algorithm:Solved nodesClosestthDistance todirectly connectedconnectedtotalnearestth nearestLastto unsolved nodes888unsolved nodedistancenodenodeconnection1SF'F'SF#SG(G(SGFH' ( œ "$$SE*E*SEFH' ( œ "$GI( ' œ "$%EH* & œ "%H"$FHFH' ( œ "$GI( ' œ "$IGI&HX"$ ' œ "*X"*HXIX"$ ( œ #!The shortest-path algorithm required additions and comparisons whereas dynamic)'programming required additions and comparisons. Hence, the latter seems to be more($efficient for shortest-path problems with "layered" network graphs.11.2-2.(a)The optimal routes are and , the associated salesS E J XS G L Xincome is 40. The route corresponds to assigning , , and "S E J X"#$salespeople to regions , , and respectively. The route corresponds" #$S G L Xto assigning , , and salespeople to regions , , and respectively.$ #"" #$