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A Graphical Algorithm for Solving an Investment Optimization Problem
Evgeny R. Gafarov Alexandre Dolgui Alexander A. Lazarev Frank Werner
Otto-von-Guericke Universität Magdeburg
Institute of Control Sciences of the Russian Academy of Sciences
Institute of Control Sciences of the Russian Academy of Sciences
Ecole Nationale Superieure des Mines
Outline
• Problem formulation;
• Dynamic programming and graphical algorithms;
• FPTAS for 6 scheduling problems.
Investment Problem
3. H. Kellerer, U. Pferschy and D. Pisinger, Knapsack Problems, Springer-
Verlag, Berlin, 2004.
4. D.X. Shaw and A. P. M. Wagelmans, An Algorithm for Single-Item Capacitated
Economic Lot Sizing with Piecewise Linear Production Costs and General Holding
Costs, Management Science, Vol. 44, No. 6, 1998, 831-838.
5. S. Kameshwaran and Y. Narahari, Nonconvex Piecewise Linear Knapsack
Problems, European Journal of Operational Research, 192, 2009, 56- 68.
Investment Problem
Running time of the classical dynamic programming algorithm: O(nA2). Running time of the best known dynamic programming algorithm: O(∑kjA).
In the graphical algorithm, the functions fj(t) and the Bellman functions (value function) Fj(t) are saved in a tabular form:
Running time of the 1st version of the graphical algorithm: O(nkmaxA log(kmaxA))
Running time of the 2nd version of the graphical algorithm: O(∑kjA)
Running time of the FPTAS based on the graphical algorithm: O(n(loglog n)∑k/ε)
Graphical Algorithms for the Investment Problem
Graphical algorithm for Investment problem
FPTAS for 6 scheduling problems
FPTAS based on the Graphical Algorithm
In the table, 0<bl
1<bl2<… since function F(t) is monotonic with t being the starting time.
The running time of the graphical algorithm is O(n min{UB,d}) for each straddling job x.
Let . Round blk up or down to the nearest multiple of
To reduce the running time, we can round (approximate) the values blk<UB to get a
polynomial number of different values blk
FPTAS based on the Graphical Algorithm
The running time of the FPTAS is
FPTAS for 6 scheduling problems
Thanks for attention
Gafarov Evgeny, Dolgui Alexandre, Lazarev Alexander, Werner Frank