Solutions_Ch11.pdf - CHAPTER 11 DYNAMIC PROGRAMMING 11.2-1(a The nodes of the network can be divided into\"layers such that the nodes in the 8th layer

Solutions_Ch11.pdf - CHAPTER 11 DYNAMIC PROGRAMMING...

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11-1 CHAPTER 11: DYNAMIC PROGRAMMING 11.2-1. (a) The nodes of the network can be divided into "layers" such that the nodes in the th 8 layer are accessible from the origin only through the nodes in the st layer. These Ð8  "Ñ layers define the stages of the problem, which can be labeled as . The nodes 8 œ "ß #ß $ß % constitute the states. Let denote the set of the nodes in the th layer of the network, i.e., , W 8 W œ ÖS× W œ 8 " # ÖEß Fß G× W œ ÖHß I× W œ ÖX× B , and . The decision variable is the immediate $ % 8 destination at stage . Then the problem can be formulated as follows: 8 0 Ð=Ñ œ Ò-  0 ÐB ÑÓ ´ 0 Ð=ß B Ñ = − W 8 œ "ß #ß $ 8 8" B −W B −W =B 8 8 8 8 for and min min 8 8" 8 8" 8 0 ÐXÑ œ ! % (b) The shortest path is . S  F  H  X (c) Number of stages: 3 6 = 0 Ð=Ñ B H X I ( X $ $ $ = 0 Ð=ß HÑ 0 Ð=ß IÑ 0 Ð=Ñ B E "" "" H F "$ "& "$ H G "$ "$ I # # # # # = 0 Ð=ß EÑ 0 Ð=ß FÑ 0 Ð=ß GÑ 0 Ð=Ñ B S #! "* #! "* F " " " " " " Optimal Solution: , and . B œ F B œ H B œ H " # $
11-2 (d) Shortest-Path Algorithm: Solved nodes Closest th Distance to directly connected connected total nearest th nearest Last to unsolved nodes 8 8 8 unsolved node distance node node connection 1 S F ' F ' SF # S G ( G ( SG F H '  ( œ "$ $ S E * E * SE F H '  ( œ "$ G I (  ' œ "$ % E H *  & œ "% H "$ FH F H '  ( œ "$ G I (  ' œ "$ I GI & H X "$  ' œ "* X "* HX I X "$  ( 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 sales S  E  J  X S  G  L  X income is 40. The route corresponds to assigning , , and " S  E  J  X " # $ salespeople to regions , , and respectively. The route corresponds " # $ S  G  L  X to assigning , , and salespeople to regions , , and respectively. $ # " " # $