ISYE3400_CH_11_DynamicProg-Spring2018.pdf

# Stage n investment n state n 1000 invested in

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Stage n = investment n ; State n = \$1000 invested in investment n If we invest \$1000 in investment 2 (i.e. state 1 of stage 2), does the amount of money that can be invested in investment 3 depend on how we got to state 1 in stage 2? Yes! We need to know state of previous stage (i.e. investment 1) If we invested \$5000 in investment 1, then \$0 available for investment 3. If we invested \$2000 in investment 1, then \$3000 available for investment 3. Therefore cannot be solved using DP as currently formulated. Redefine stage and/or state DP Example 2 - Resource Allocation Problem 20

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Ø Often tricky to formulate as DP. A certain degree of ingenuity and insight into the general structure of dynamic programming problems is required to recognize when and how a problem can be solved by dynamic programming procedures. Ø Not all problems can be solved using DP DP Example 2 - Resource Allocation Problem 21
Stage n = investments n thru 3 {1,2,3} State n = available money for investments n thru 3 {0,1,2,3,4,5,6} Arc length = NPV 22 Stage 2 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 Stage 3 Stage 4 Stage 1 6 \$ to investment 1 \$ to investment 2 \$ to investment 3 \$1000 available for investment 3 \$2000 available for investments 2 & 3 DP Example 2 - Resource Allocation Problem We want to maximize NPV, so find LONGEST path

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First check if this can be solved using DP Does the correct decision at any stage depend on previous stages? Stage n = investments n thru 3 State n = available money for investments n thru 3 If we invest \$4000 in investments 1 & 2 (i.e. state 2 of stage 3), does the amount of money that can be invested in investment 3 depend on how we got to state 2 in stage 3? No! Amount of available money for investment 3 is \$2000 Doesn’t matter whether we spent \$1000 on investment 1 and \$3000 on investment 2 OR \$0 on investment 1 and \$4000 on investment 2, we still have \$2000 for investment 3. Therefore can be solved using DP as currently formulated. 23 DP Example 2 - Resource Allocation Problem
24 We want to maximize NPV, so find LONGEST path Note arcs out of stage 2 e.g. cannot have more than \$2000 available in stage 3 Stage 2 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 Stage 3 Stage 4 Stage 1 6 DP Example 2 - Resource Allocation Problem Stage n = investments n thru 3 {1,2,3} State n = available money for investments n thru 3 {0,1,2,3,4,5,6} Arc length = NPV

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Use backward recursion - start at “end” of problem and work forward. Stage 3 Calculations Find longest path (i.e. largest NPV) for each \$1000 invested in investment 3 f 3 (0) = NPV 3 (0) = 0 f 3 (1) = NPV 3 (1) = 4(1) + 5 = 9 f 3 (2) = NPV 3 (2) = 4(2) + 5 = 13 f 3 (3) = NPV 3 (3) = 4(3) + 5 = 17 f 3 (4) = NPV 3 (4) = 4(4) + 5 = 21 f 3 (5) = NPV 3 (5) = 4(5) + 5 = 25 f 3 (6) = NPV 3 (6) = 4(6) + 5 = 29 ← \$0 available for investment 3. ← \$4000 available for investment 3. Notation: f n (i) = length of longest path from state i in stage n to end DP Example 2 - Resource Allocation Problem
Stage 2 Calculations Find longest path (i.e. largest NPV) for each \$1000 invested in investments 2 and 3. Calculated as function of stage 3 longest paths.

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• Spring '10
• CHEN
• Shortest path problem, Longest path problem, DP Applications

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