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