1. A Graphical Approach for Solving Single Machine Scheduling Problems Approximately 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 Ecole Nationale Superieure des Mines

2. Outline of the Talk 1. Dynamic Programming and Graphical Algorithms 2. An FPTAS based on the Graphical Algorithm 3. An Overview of Graphical Algorithms for Single Machine Problems 4. Graphical Algorithms for an Investment Problem

3. Single machine n jobs j = 1,2,…,n p j processing time d j =d common due date w j weight Tardiness of job j in schedule π : T j (π) = max{0,C j (π)-d} Goal: Find a schedule π* that minimizes ∑w j T j Dynamic Programming Algorithms for the Problem 1|d j =d|∑w j T j

4. 4 Lemma 1: There exists an optimal schedule π = (G,x,H), where all jobs from set G are on-time and processed in non-increasing order of the values p j /w j ; all jobs from set H are tardy and processed in non-decreasing order of the values p j /w j ; the straddling job x starts before time d and is completed no earlier than time d. Lemma 1: There exists an optimal schedule π = (G,x,H), where all jobs from set G are on-time and processed in non-increasing order of the values p j /w j ; all jobs from set H are tardy and processed in non-decreasing order of the values p j /w j ; the straddling job x starts before time d and is completed no earlier than time d. Dynamic Programming Algorithms for the Problem 1|d j =d|∑w j T j

5. First Dynamic Programming Algorithm for the Problem 1|d j =d|∑w j T j l t π l-1 (t+p l ) Let x=1 be the straddling job. In step l, l = 1,2,…, n for each state t=[0, ∑p j ] or [0,d] we choose one of two positions for job l: l t π l -1 (t) The running time is O(nd) for each straddling job x=1,2,…,n

6. Graphical Algorithm Dynamic Programming (Bellman 1954) Idea of the graphical algorithm: Combine several states into a new state

7. Computations in the dynamic programming algorithm Computations in the graphical algorithm Graphical Algorithm

9. f j+1 = min{Ф 1,Ф 2 }

10. Graphical Algorithm In the table, 0<b l 1 <b l 2 <… since function F(t) is monotonic with t being the starting time. Function F l (t) can be defined for all t from (-∞,+∞). Let UB be an upper bound on the optimal objective function value. Then we have to save only the columns with b l k <UB. The running time of the Graphical Algorithm is O(n min{UB,d}) for each straddling job x.

11. FPTAS based on the Graphical Algorithm In the table, 0<b l 1 <b l 2 <… 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 b l k up or down to the nearest multiple of To reduce the running time, we can round (approximate) the values b l k <UB to get a polynomial number of different values b l k

12. FPTAS based on the Graphical Algorithm The running time of the FPTAS is

13. Graphical Algorithms and the corresponding FPTAS

14. n investment projects А – investment budget (for all A from the interval [A’,A’’]) f j (t) -- profit function of project j The goal is to define an amount t j in [0,A] (integer) for each project to maximize the total profit. ∑ t j <= A t j is integer Investment Problem

16. Classical dynamic programming algorithm: O(nA 2 ) running time. The best known dynamic programming algorithm: O(∑k j A) running time. In the Graphical Algorithm functions f j (t) and the Bellman functions (value function) F j (t) are saved in a tabular form: Running time of the 1 st version of the graphical algorithm: O(nk max A log(k max A)) Running time of the 2 nd version of the graphical algorithm: O(∑k j A) Running time of the FPTAS based on the graphical algorithm: O(n(loglog n)∑k/ε) Graphical Algorithms for the Investment Problem

17. Thanks for your attention 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 Ecole Nationale Superieure des Mines