1. Generating RCPSP instances with Known Optimal Solutions José Coelho jcoelho@univ-ab.pt Generator and generated instances in: http://jcoelho.m6.net/edicao3.asp?pa=3569

2. Generating RCPSP instances with Known Optimal Solutions Index 1.Introduction 2.Generation Method 3.Tests 4.Conclusions

3. Generating RCPSP instances with Known Optimal Solutions 1.Introduction  There are several algorithms for solving the RCPSP  The comparison of algorithms requires an instance set  The average performance is extrapolated to all RCPSP instances  A diversified instance set minimizes extrapolation errors  To generate a diversified instance set, one needs to Identify the relevant indicators However these indicators are not consensual  There are several generators, with different kinds of arguments  The resources are generated randomly, in such a way that resource indicators are verified, but not getting the optimal solution  The most widely used instance set is PSPLIB

4. Generating RCPSP instances with Known Optimal Solutions 1.Introduction  Available generators do not give any optimal solution RCPSP belongs to NP-hard  It is impossible to calculate an optimal solution for hard instances  For PSPLIB, only the J30 subset has optimal solutions for all instances Without optimal solutions  The comparisons need to be relative to the best lower or upper bound  Or relative to a predefined lower bound (CPM-value) With optimal solutions  The calculation of the exact performance of algorithms is possible  The study of instance's complexity will have a more solid base  The proposed generator GenRes: Uses a network and desired resource indicators RF/RC Generates resource information Returns a RCPSP instance with an optimal solution

5. Generating RCPSP instances with Known Optimal Solutions 2. Generation Method  Generation arguments A Network K extra precedence relations SR saturated resources Optional (default value is read from input instance)  Number R of resources  Desired resource indicators RF/RC  Desired sum of all processing times  Phases 1. Extra precedence relations 2. Resource usage 3. Processing times

6. Generating RCPSP instances with Known Optimal Solutions 2. Generation Method Phase 1. Extra precedence relations  Set processing times to 1  Calculate the earliest start schedule (ESS)  Add K extra precedence relations Select at random two activities A, B, to add a precedence relation Accept precedence relation from A to B if  Adding the precedence relation change the ESS  Total processing time does not exceed L: If precedence relation is accepted, update the ESS, otherwise try another two activities

7. Generating RCPSP instances with Known Optimal Solutions 2. Generation Method Phase 2. Resource Usage  For the first SR saturated resources Arrange activities in random order Set unary use of resource for the first activities with different start instants Set unary use of resource for the first activities that does not use the resource, until RF is attained  For other non saturated resources Arrange activities in random order Set unary use of resource for the first activities, until RF is attained  Until the resource is saturated, or RC is archived Select at random an activity with a resource usage Increase resource usage of activity if resource usage in activity start instant is less than resource capacity

8. Generating RCPSP instances with Known Optimal Solutions 2. Generation Method Phase 3. Processing Times  Repeat until the sum of all processing times is attained Select a start instance at random Increase processing times of activities that start at that instant  Calculate the ESS and save it as an optimal schedule  Discard the extra precedence relations and return the original network with generated resources

9. Generating RCPSP instances with Known Optimal Solutions 3. Tests  Questions: I.Is GenRes capable of generating instances of all types? II.Does the complexity of generated instances go from easy to hard? III.What is the influence of generator arguments, SR, K and R, on the results?  The GenRes was tested using PSPLIB instances as argument The K used is 100, and SR is set to 1 Optional arguments R/RF/RC are not set, the generator will try to match instance values

10. Generating RCPSP instances with Known Optimal Solutions 3. Tests I. Is GenRes capable of generating instances of all types?  Figures A - RF (blue) and RC (green) original values of PSPLIB, versus values of generated instances B - RF versus RC of original (blue) and generated (green)  Comments For all instances, original RF and RC values are accomplished The answer is yes, if RF and RC cover all types of resource instances

11. Generating RCPSP instances with Known Optimal Solutions 3. Tests II. Does the complexity of generated instances go from easy to hard?  Figures C - parallel versus serial scheduling of LST in PSPLIB D - serial LST rule value, relative to the best upper bound (blue), best lower bound (green), and to the optimal solution (red) for generated instances  Comments Using lower or upper bound may lead to very different conclusions About half of PSPLIB instances are easy Generated instances are more equally distributed from easy to hard

12. Generating RCPSP instances with Known Optimal Solutions 3. Tests III. What is the influence of generator argument SR on the results? (1/3)  Figure E - performance of serial LST rule for the generated resources with SR equal to 1 (blue), 2 (green) and 3 (red)  Comments The average complexity of instances increase when SR increase There are always some easy instances

13. Generating RCPSP instances with Known Optimal Solutions 3. Tests III. What is the influence of generator argument K on the results? (2/3)  Figures F - LST rule for K from 100 to 4 G - LST rule for K equal to 100, 2 and 1  Comments High value for K does not make much difference Value 1 or 2 for K increase the number of easy instances The number of hard instances does not decrease very much even with K=1

14. Generating RCPSP instances with Known Optimal Solutions 3. Tests III. What is the influence of generator argument R on the results? (3/3)  Figures H - LST rule for R from 2 to 16 I - scatter plot of R=2 vs R=4 and R=8 vs R=16  Comments Increasing R makes the number of instances of average difficulty decrease and the number of hard instances increase An instance that is hard with R=8 is hard with R=16

15. Generating RCPSP instances with Known Optimal Solutions 4. Conclusions  The GenRes generator that returns an optimal solution was presented Lower and upper bounds can be very different in hard instances  The generator can produce instances diverse in RF/RC  The distribution of instance complexity is well distributed  Increasing SR and R increases the instance hardness/difficulty  K appears to have no effect on the complexity, except for very low values

16. Generating RCPSP instances with Known Optimal Solutions 4. Future Work  Research on instance complexity What makes an instance hard? Is it possible to explain the instance hardness with a set of indicators? Has morphology something to do with it?  Generation of an instance set Diverse not only in RF/RC but also in R and SR Diverse in all indicators related with complexity  Comparing algorithms Average performance of an algorithm Average performance of worst 5% instances of an algorithm (some type of worst case analysis)

17. Generating RCPSP instances with Known Optimal Solutions José Coelho jcoelho@univ-ab.pt Generator and generated instances in: http://jcoelho.m6.net/edicao3.asp?pa=3569