THE STABILITY BOX IN INTERVAL DATA FOR MINIMIZING THE SUM OF WEIGHTED COMPLETION TIMES Yuri N. Sotskov Natalja G. Egorova United Institute of Informatics Problems, National Academy of Sciences of Belarus Tsung-Chyan Lai Department of Business Administration, National Taiwan University Frank Werner Faculty of Mathematics, Otto-von-Guericke-University

2 Outline of the Talk 1. Introduction 2. Problem Setting 3. Stability Box 4. Computational Results 5. Conclusion and Further Work

3 1. Introduction Approaches dealing with scheduling problems under uncertainty 1.Stochastic method (Pinedo, 2002) 2.Fuzzy method (Slowinski and Hapke, 1999) 3.Robust method (Daniels and Kouvelis, 1995; Kasperski, 2005; Kasperski and Zelinski, 2008) 4.Stability method (Lai and Sotskov, 1999; Lai et al., 1997; Sotskov et al., 2009; Sotskov et al., 2010a; Sotskov et al., 2010b)

4 2. Problem setting - set of n jobs - weight for a job - processing time of a job, - set of vectors of the possible job processing times Find an optimal permutation such that Problem,

Problem is not correct. The deterministic problem is correct and can be solved exactly in O(n log n) time (Smith, 1956). Necessary and sufficient condition for the optimality of a permutation Deterministic case: for each job 5

6 3. Stability box - set of permutations - permutation of the jobs - subset of set - set of permutations with the largest dimension and volume of the stability box

7 Definition 1. (Sotskov and Lai, 2011) The maximal closed rectangular box. is a stability box of permutation, if permutation. being optimal for the instance with a scenario remains optimal for the instance. with a scenario. for each. If there does not exist a scenario such that permutation is optimal for the instance, then.

Precedence-dominance relation on the set of jobs J and the solution concept of a minimal dominant set The set of permutations is a minimal dominant set for a problem, if for any fixed scenario, the set S(T) contains at least one optimal permutation for the instance,, provided that any proper subset of set S(T) loses such a property. Definition 2. (Sotskov and al., 2009)

9 Theorem 1. (Sotskov and al., 2009) For the problem, job dominates job if and only if the following inequality holds: Definition 3. (Sotskov and al., 2009) Job dominates job, if there exists a minimal dominant set S(T) for the problem such that job precedes job in every permutation of the set S(T).

10 Lower (upper) bound ( ),, on the maximal range of possible variations of the weight-to-process ratio preserving the optimality of permutation : 3.2. Calculating the stability box,,

11 If there is no job,, in permutation such that inequality holds for at least one job,, then the stability box is calculated as follows: Otherwise, Theorem 2. (Sotskov and Lai, 2011)

Illustrative example

13 Stability box for Relative volume of a stability box

Properties of a stability box Property 1. For any jobs and, Case (I) For case (I), there exists a permutation, in which job proceeds job. Property 2.

15 Property 3. For case (II), there exists a permutation, in which jobs and are located adjacently: and. Remark 1. Due to Property 3, while looking for a permutation, we shall treat a pair of jobs satisfying (1) as one job (either job or ). Case (II) (1)

Case (III) Property 4. (i) For a fixed permutation, job may have at most one maximal segment of possible variations of the processing time preserving the optimality of permutation. (ii) For the whole set of permutations S, only in case (III), a job may have more than one (namely: ) maximal segments of possible variations of the time preserving the optimality of this or that particular permutation from the set S. (2) 16

17 Property 5. L - the set of all maximal segments of possible variations of the processing times for all jobs preserving the optimality of permutation. Property 6. There exists a permutation with the set of maximal segments of possible variations of the processing time, preserving the optimality of permutation.

A job permutation with the largest volume of a stability box Algorithm MAX-STABOX Input: Segments, weights,. Output: Permutation, stability box. Step 1: Construct the list and the list in non-decreasing order of. Ties are broken via increasing. Step 2: Construct the list and the list in non-decreasing order of. Ties are broken via increasing.

19 Step 3: FOR j = 1 to j = n DO compare job and job. Step 4: IF THEN job has to be located in position j in permutation. GOTO step 8. Step 5: ELSE job satisfies inequalities (2). Construct the set of all jobs satisfying inequalities (2), where Step 6: Choose the largest range among those generated for job. Step 7: Partition the set J(i) into subsets and generating the largest range. Set j = k+1 GOTO step 4. Step 8: Set j := j+1 GOTO step 4. END FOR Step 9: Construct the permutation via putting the jobs J in the positions defined in steps 3 – 8. Step 10: Construct the stability box. STOP.

20 Remark 2. Algorithm MAX-STABOX constructs a permutation such that the dimension of the stability box is the largest one for all permutations S, and the volume of the stability box is the largest one for all permutations having the largest dimension of their stability boxes.

21 4. Computational results C - integer center of a segment was generated using the uniform distribution in the range [L;U]:. - maximal possible error of the random processing times - lower (upper) bound for the possible job processing time, - the average relative number of the arcs in the dominance digraph - relative error of the objective function value - the optimal objective function value of the actual scenario Each series contains 100 solved instances Processor AMD Athlon (tm) , 2.00 GHz; RAM 1.96 GB

22

23 5. Conclusion and further work Further research concerning the use of a stability box for other uncertain scheduling problems appear to be promising.  An algorithm has been developed for calculating a permutation. with the largest dimension and volume of a stability box.. ;  Properties of a stability box were proved allowing to derive an O(nlogn) algorithm for calculating a permutation ;  The dimension and volume of a stability box are efficient invariants of uncertain data T, as it is shown in simulation experiments on a PC.

24 6. References 1.Lai, T.-C., Sotskov, Y., Sotskova, N., and Werner, F. (1997). Optimal makespan scheduling with given bounds of processing times. Mathematical and Computer Modelling, V. 26(3):67–86. 2.Smith, W. (1956). Various optimizers for single-stage production. Naval Research Logistics Quarterly, V. 3(1):59–66. 3.Sotskov, Y., Egorova, N., and Lai, T.-C. (2009). Minimizing total weighted flow time of a set of jobs with interval processing times. Mathematical and Computer Modelling, V. 50:556– Sotskov, Y., Egorova, N., and Werner, F. (2010a). Minimizing total weighted completion time with uncertain data: A stability approach. Automation and Remote Control, V. 71(10):2038– Sotskov, Y. and Lai, T.-C. (2011). Minimizing total weighted flow time under uncertainty using dominance and a stability box. Computers & Operations Research. doi: /j.cor Sotskov, Y., Sotskova, N., Lai, T.-C., and Werner, F. (2010b). Scheduling under Uncertainty. Theory and Algorithms. Belorusskaya nauka, Minsk, Belarus. 7.Sotskov, Y., Wagelmans, A., and Werner, F. (1998). On the calculation of the stability radius of an optimal or an approximate schedule. Annals of Operations Research, V. 83:213–252.