Combining Iterated Local Search and Biased Randomization for solving non-smooth flow-shop problems

1. IEMAE Combining Iterated Local Search and Biased Randomization for solving non-smooth flow-shop problems Albert Ferrer, Angel A. Juan, Helena R. Lourenço alberto.ferrer@upc.edu Dep. Applied Mathematics I Universitat Politécnica de Catalunya, SPAIN

2. Overview • Convex Optimization Problems (COPs) • Non-Convex Optimization Problems (NCOPs) • Non-Smooth Optimization Problems (NSPs) • The Flow-Shop problem (Makespan) • The NEH heuristic for FSP • The ILS-ESP algorithm for FSP • Randomizing Classical Heuristics • A More Realistic Modelfor FSP (Tardiness) • Conclusions and Future Work

3. Overview • Convex Optimization Problems (COPs) • Non-Convex Optimization Problems (NCOPs) • Non-Smooth Optimization Problems (NSPs) • The Flow-Shop problem (Makespan) • The NEH heuristic for FSP • The ILS-ESP algorithm for FSP • Randomizing Classical Heuristics • A More Realistic Modelfor FSP (Tardiness) • Conclusions and Future Work

4. Overview • Convex Optimization Problems (COPs) • Non-Convex Optimization Problems (NCOPs) • Non-Smooth Optimization Problems (NSPs) • The Flow-Shop problem (Makespan) • The NEH heuristic for FSP • The ILS-ESP algorithm for FSP • Randomizing Classical Heuristics • A More Realistic Modelfor FSP (Tardiness) • Conclusions and Future Work

5. Overview • Convex Optimization Problems (COPs) • Non-Convex Optimization Problems (NCOPs) • Non-Smooth Optimization Problems (NSPs) • The Flow-Shop problem (Makespan) • The NEH heuristic for FSP • The ILS-ESP algorithm for FSP • Randomizing Classical Heuristics • A More Realistic Modelfor FSP (Tardiness) • Conclusions and Future Work

6. 1. Convex Optimization Problems (COPs) A convex function • COPs are problems where all of the constraints are convex functions, and the objective is a convex function if minimizing, or a concave function if maximizing. • Linear functions are convex, so LP problems are COPs. • In a COP, the feasible region –the intersection of convex constraint functions– is also a convex region. LPs are COPs • With a convex objective and a convex feasible region, eachoptimal solutionis globally optimal. Several methods –e.g. Interior Point methods– will either find the globally optimal solution, or prove that there is no feasible solution to the problem. • COPs can be solved efficiently up to very large size. A convex region Source: www.solver.com

7. 2. Non-convex Optimization Problems (NCOPs) A non-convex function with multiple local minima • NCOPs are problems where either the objective or any of the constraints are non-convex. • NCOPs may have multiple feasible regions and multiple locally optimal points within each region. • It can take exponential time in the number of variables and constraints to determine that a non-convex problem is infeasible, that the objective function is unbounded, or that an optimal solution is the "global optimum" across all feasible regions. Non-convex regions Source: www.solver.com

8. 3. Non-Smooth Optimization Problems (NSPs) • Typically, NSPs are also NCOPs. Hence: • They might have multiple feasible regions and multiple locally optimal points within each region –because some of the functions are non-smooth or even discontinuous, and • Derivative/gradient information generally cannot be used to determine the direction in which the function is increasing (or decreasing). • In a NSP, the situation at one possible solution gives very little information about where to look for a better solution. Examples of non-smooth functions • In most NSPs it is impractical to enumerate all of the possible solutions and obtain the best one. Hence, most methods rely on some sort of random sampling of possible solutions. • Such methods are nondeterministic or stochastic –they may yield different solutions on different runs, depending on which points are randomly sampled. Source: www.mathworks.com

9. 3. Examples of Nonsmooth Functions The Problem The objective function • Sensor Network Localization • Optimal Circuit Routing • Winner Determination Problem

10. 4. The Flow-Shop problem (1/2) • The Permutation Flowshop Sequencing Problem (PFSP) is a well-known scheduling problem that can be described as follows: • A set J of n independent jobs has to be processed on a set M of m independent machines. • Every job j in J requires a given fixed processing time pij > 0 on every machine i in M. • Each machine can execute at most one job at a time. • All jobs are processed by the machines in the same order. Jobs are processed by machines Goal: To find the sequence of jobs that minimizes the makespan • Classical goal:to find a sequence for processing the jobs in the shop so that the completion time or makespan is minimized.

11. 4. The Flow-Shop problem (2/2) • The PFSP is a combinatorial problem with n! possible sequences which, in general, is NP-complete. • A large number of different approaches have been developed: from pure optimization methods (e.g. mixed integer programming) to the use of heuristics and metaheuristics. Metaheuristics • Still, there is a need for new efficient, simple and parameter-free (ESP) methods able to provide a large set of alternative near-optimal solutions with different properties, so that decision-makers can choose among different alternative solutions according to their specific necessities and preferences. Utility function

12. For each job, calculate its total processing time Job i Job 1 Job 2 … Construct a sorted efficiency list (jobs with higher processing times at the top) Job n Job 3 Each new job is inserted into the “best” possible position Job 1 Job 2 5. The NEH heuristic (Nawaz et al. 1983) • Calculate the total processing time required for each job, and then to create an “efficiency list” of jobs sorted in a descending order NEH is probably the most efficient heuristic for the PFSP • At each step, the job at the top of the efficiency list is selected and used to construct the solution. • Once selected, a job is inserted in the sorted set of jobs that are configuring the ongoing solution. The exact position that the selected job will occupy in that ongoing solution is given by the minimizing makespan criterion. At each iteration, select the job at the top of the list

13. 6. The ILS-ESP algorithm • The ILS-ESP is based on the IG-ILS (Ruiz 2007). Itis equally Efficient, Simpler and Parameter-free. • It incorporates 3 major changes with respect to IG-ILS: (1) the RandNEH method, which provides diversification in the initial solution, (2) a new ‘enhanced swap’ operator, and (3) a simpler Demon-based acceptance criterion. ILS-ESP is an efficient, simple and parameter-free algorithm for the PFSP Iterated Local Search framework 0 parameters Randomized NEH solution Enhanced Swap operator Demon-like acceptance criterion • We have tested ILS-ESP vs. IG-ILS using the Taillard benchmarks (120 instances) and they seem to offer equivalent performance: Average GAP = 0.36% in 10ms*nJobs*nMachines and after running each algorithm 10 times (replicas).

14. 7. Randomizing Classical Heuristics • We propose to introduce a biased random behavior (BRB) in the selection processes of classical heuristics so that movements with better values have higher probabilities of being selected, but other movements could also be selected instead at each step. • This way, deterministic classical heuristics (e.g.: Clarke and Wright, NEH, etc.) are transformed into probabilistic ones without losing the “common sense” rules that make them efficient. Geometric distributions • Thus, we transform a “gun heuristic” into a “machine-gun heuristic”: each time the randomized heuristic is run, a different “good” solution will be obtained (kind of a “Biased GRASP”). • The geometric and the discrete version of the triangular can be used to infer this BRB. Triangular distributions

15. 8. RandNEH: Randomizing the NEH for the PFSP • NEH  jobs are ordered in decreasing order according to their total completion time on all the machines • RandNEH introduces randomness in this process by using a triangular statistical distribution  jobs that take longer to complete will be more likely to be selected first, but all jobs in the list are potentially eligible. • Notice: Each time RandNEH is run, a different random solution is obtained. By construction, chances are that this solution outperforms the NEH one and, in any case, this solution will be a good alternative to the original NEH diversification of the starting base solution. • Generating triangular random variates • Calculate u ~ U(0,1) • Calculate X = b * (1 – Sqrt(1 – u)) • Calculate pos = Floor(X)

16. New Enhanced Swap operator: Select two positions at random. Interchange jobs at the selected positions. Perform a sorted ‘shift-to-left’ movement to ‘adjust’ the positions of the swapped jobs. New acceptance criterion: If the new solution improves the best solution, then update base solution (improvement). Else, accept the new solution as the base solution (degeneration) as far as the last movement was an improvement and the degeneration is not greater than the last improvement. 9. The Enhanced Swap & the Acceptance Criterion

17. 10. The Flow-Shop Problem: Some results Test: 15 runs per instance with maxTime = 0.010s * nJobs * nMachines Computer: Intel Xeon 2.0GHz 4GB RAM Note: All algorithms have been implemented in Java (non-optimized code) Juan, A.; Ruiz, R.; Mateo, M.; Lourenço, H.; Ionescu, D. (2010): “A Simulation-based Approach for Solving the Flowshop Problem”. In Proceedings of the 2010 Winter Simulation Conference.

18. 11. FSP: A More Realistic Model The following notation is used for the description of the flow-shop problem:

19. 12. FSP: Objective Function

20. 13. FSP: The Model

21. 14. FSP in production industries Most of the flow-shop problems arise in production industries, wherea set of products or jobs must be produced.

22. 14. FSP in production industries Most of the flow-shop problems arise in production industries, wherea set of products or jobs must be produced. These industries usually are integrated in a supply chain, meaning that after being produced they are transported to a warehouse or customers.

23. 14. FSP in production industries Most of the flow-shop problems arise in production industries, wherea set of products or jobs must be produced. These industries usually are integrated in a supply chain, meaning that after being produced they are transported to a warehouse or customers. Usually this transportation is set in advanceand it has a periodicity, for example every 2 hours or morning and afternoon.

24. 14. FSP in production industries Most of the flow-shop problems arise in production industries, wherea set of products or jobs must be produced. These industries usually are integrated in a supply chain, meaning that after being produced they are transportedto a warehouse or customers. Usually this transportation is set in advanceand it has a periodicity, for example every 2 hours or morning and afternoon. In this case the due date is related with this transportation schedule, a job is produced on time if it can meet its transportation due date, and if it late the jobs must be scheduled to later transport.

25. 15. FSP transportation scheduled • The commercial agents negotiate the date, d’j, that the job must be delivered to the customer's location (delivery date ).

26. 15. FSP transportation scheduled • The commercial agents negotiate the date, d’j, that the job must be delivered to the customer's location (delivery date ). • Assuming that we have an estimated transportation time, ξj , between the workshop and the customer, we can calculate the due date at the workshop, dj = d’j - ξj. This assumes that exist a constant transportation to the customer which is not usually the case.

27. 15. FSP transportation scheduled • The commercial agents negotiate the date, d’j, that the job must be delivered to the customer's location (delivery date ). • Assuming that we have an estimated transportation time, ξj , between the workshop and the customer, we can calculate the due date at the workshop, dj = d’j - ξj. This assumes that exist a constant transportation to the customer which is not usually the case. • We assume that the transportation is scheduled every period of time, τ, that is there are an available transportation at the times t0, t1, t2, … , with τ = tk-tk-1

28. 15. FSP transportation scheduled • The commercial agents negotiate the date, d’j, that the job must be delivered to the customer's location (delivery date ). • Assuming that we have an estimated transportation time, ξj , between the workshop and the customer, we can calculate the due date at the workshop, dj = d’j - ξj. This assumes that exist a constant transportation to the customer which is not usually the case. • We assume that the transportation is scheduled every period of time, τ, that is there are an available transportation at the times t0, t1, t2, … , with τ = tk-tk-1 • Therefore, the job must be produced before the corresponding transportation schedule time, tkj, which is the closest tk before the workshop due date dj, tkj= max{ tk : tk <= dj}, otherwise the job can not arrive on time.

29. 16. FSP penalizations Different situations can occur:

30. 16. FSP penalizations Different situations can occur: Thejobiscompletedon time, justbeforethecorrespondenttransportationschedule, tkj - τ < Cm,j <= tkj, afterbeingproducedthejobsgoesdirectlytothevehicle, and itison time to me deliveredtothecustomer; so, thereisno penalizationforbeingearlyor late.. Thejob is completedtooearly, i.e., beforethepreviouscorrespondenttransportationschedule, Cm,j <= tkj - τ, thereforethe company has to storethisjob on a warehouse so thereexist a penalizationrelatedwiththetimethattheproduct or jobmust be stored in theworkshop. This cost makes sense, sincetoday most of theproductioncompanieshavelimitedstoringspace. Thejob is completedtoolate, i.e. afterthecorrespondenttransportationschedule, Cm,j > tkj, thereforethejob miss itscorrespondingtransportationanditwill be deliveredlate. In thiscasewepropose to types of penalization, bythetimethatthejob is lateandbythenumber of periodsthatit is late.

31. 16. FSP penalizations Different situations can occur: Thejobiscompletedon time, justbeforethecorrespondenttransportationschedule, tkj - τ < Cm,j <= tkj, afterbeingproducedthejobsgoesdirectlytothevehicle, and itison time to me deliveredtothecustomer; so, thereisno penalizationforbeingearlyor late.. Thejob is completedtooearly, i.e., beforethepreviouscorrespondenttransportationschedule, Cm,j <= tkj - τ, thereforethe company has to storethisjob on a warehouse so thereexist a penalizationrelatedwiththetimethattheproduct or jobmust be stored in theworkshop. This cost makes sense, sincetoday most of theproductioncompanieshavelimitedstoringspace. Thejob is completedtoolate, i.e. afterthecorrespondenttransportationschedule, Cm,j > tkj, thereforethejob miss itscorrespondingtransportationanditwill be deliveredlate. In thiscasewepropose to types of penalization, bythetimethatthejob is lateandbythenumber of periodsthatit is late.

32. 16. FSP penalizations Different situations can occur: Thejobiscompletedon time, justbeforethecorrespondenttransportationschedule, tkj - τ < Cm,j <= tkj, afterbeingproducedthejobsgoesdirectlytothevehicle, and itison time to me deliveredtothecustomer; so, thereisno penalizationforbeingearlyor late.. Thejob is completedtooearly, i.e., beforethepreviouscorrespondenttransportationschedule, Cm,j <= tkj - τ, thereforethe company has to storethisjob on a warehouse so thereexist a penalizationrelatedwiththetimethattheproduct or jobmust be stored in theworkshop. This cost makes sense, sincetoday most of theproductioncompanieshavelimitedstoringspace. Thejob is completedtoolate, i.e. afterthecorrespondenttransportationschedule, Cm,j > tkj, thereforethejob miss itscorrespondingtransportationanditwill be deliveredlate. In thiscasewe proposetwotypes of penalization, bythe time thatthejob is late and bythenumber of periodsthatit is late.

33. 17. FSP: Earliness and Tardiness If a job is produced to early and must be stored to wait for the corresponding transportation we have the following cost: If a job is produced to late, it misses the corresponding transportation and it is delivered late, the penalization cost is as follows: In the case of a job being late, it is common practice to penalize by the number of period that the job is late since if the job misses its transportation must wait for the next one.

34. 18. Conclusions & Future Work • We have discussed the use of probabilistic or stochastic algorithms for solving non-smooth combinatorial optimization problems in the case of FSP.

35. 18. Conclusions & Future Work • We have discussed the use of probabilistic or stochastic algorithms for solving non-smooth combinatorial optimization problems in the case of FSP. • We propose the use of probability distributions, such as the Triangular one, to add a biased random behavior to classical NEH heuristic for the Flow Shop Scheduling Problem.

36. 18. Conclusions & Future Work • We have discussed the use of probabilistic or stochastic algorithms for solving non-smooth combinatorial optimization problems in the case of FSP. • We propose the use of probability distributions, such as the Triangular one, to add a biased random behavior to classical NEH heuristic for the Flow Shop Scheduling Problem. • By randomizing these heuristic, a large set of alternative good solutions can be quickly obtained in a natural and easy way.

37. 18. Conclusions & Future Work • We have discussed the use of probabilistic or stochastic algorithms for solving non-smooth combinatorial optimization problems in the case of FSP. • We propose the use of probability distributions, such as the Triangular one, to add a biased random behavior to classical NEH heuristic for the Flow Shop Scheduling Problem. • By randomizing these heuristic, a large set of alternative good solutions can be quickly obtained in a natural and easy way. • Some specific example in the makespan version of this technique have been analyzed to illustrate the main ideas behind this approach.

38. 18. Conclusions & Future Work • We have discussed the use of probabilistic or stochastic algorithms for solving non-smooth combinatorial optimization problems in the case of FSP. • We propose the use of probability distributions, such as the Triangular one, to add a biased random behavior to classical NEH heuristic for the Flow Shop Scheduling Problem. • By randomizing these heuristic, a large set of alternative good solutions can be quickly obtained in a natural and easy way. • Some specific example in the makespan version of this technique have been analyzed to illustrate the main ideas behind this approach. • An extension of the FSP that includes intermediary inventory holding cost and the sum of earliness and tardiness has been described in order to present a more realistic model (in progress).

39. IEMAE Combining Iterated Local Search and Biased Randomization for solving non-smooth flow-shop problems Thank You! Albert Ferrer, Angel A. Juan, Helena R. Lourenço alberto.ferrer@upc.edu Dep. Applied Mathematics I Universitat Politécnica de Catalunya, SPAIN