# Permutation flowshop scheduling by genetic local search

## Summary (3 min read)

### 1 Introduction

- It is well known that Genetic Algorithms (GAs) can be enhanced by incorporating constructive heuristics or pointbased local search methods.
- Section 3 explains the GLS approach based on the stochastic local search and MSXF.
- Section 4 investigates the existence of a ‘big valley structure’ for the PFSP with stochastic local search and the representative neighbourhood.
- Experimental results demonstrate the effectiveness of the proposed method.

### 2 The permutation flowshop scheduling problem

- The permutation flowshop scheduling problem (PFSP) is often designated by the symbolsn=m=P=Cmax, wheren jobs have to be processed onmmachines in the same order.
- P indicates that only permutation schedules are considered, where the order in which each machine processes the jobs is identical for all machines.
- More sophisticated operators are obtained by limiting the size of the neighbourhood by using the notion of critical blocks.
- For each jobin a critical block, letSaj be a set of moves that shift the jobj to some position in the next block; similarlySbj shiftsj to the previous block.
- In this paper two well-known distances are considered as follows: precedence-based distance:.

### 3 Genetic local search

- It is well known that GAs are not well suited for finetuning structures that are very close to optimal solutions, and that it is essential to incorporate local search methods, such as neighbourhood search, into GAs.
- The result of such incorporation is often calledGenetic Local Search(GLS) [10].
- This approach can be viewed as a variant of Adaptive Multi-Start (AMS) methods in which local search is applied repeatedly, each time a new starting point being generated adaptively based on previously found local optima [2].
- The Multi-Step Crossover Fusion (MSXF) GA proposed by Yamada and Nakano [12] is one such GLS method, and it has been applied successfully to job-shop scheduling problems.
- This section briefly reviews neighbourhood search and the MSXF.

### 3.1 Neighbourhood search

- An outline of a neighbourhood search (NS) for minimizing V (x) is described in Algorithm 3.1, wherex denotes a point in the search space,V (x) denotes its objective function value andN(x) its neighbourhood.
- The termination condition can be given, for example, as a fixed number of iterationsL.
- Step 1 in Algorithm 3.1 defines the NS operator: the main part of NS.
- This operator is categorized by the way a point is selected fromN(x), which is called thechoice criterion.
- Simulated Annealing (SA) is a method in which the parameterT (called thetemperature) decreases to zero following an annealing schedule as the number of iterations increases.

### 3.2 Multi-step crossover fusion

- The genetic crossover operator has two functions, which the authors denote by F1 and F2.
- MSXF carries out a short term ‘navigated’ local search starting from one of the parent solutions to find new good solutions (F2), where the other parent is used as a reference point so that the search direction is biased toward it and therefore the search is limited between the parents (F1).
- Let the parent solutions bep0 andp1, and let the distance between any two individualsx andy in any representation be d(x; y).
- Hered(yi; p1) can be estimated easily if d(x; p1) and the direction of the transition fromx to yi are Algorithm 3.2 Multi-Step Crossover Fusion (MSXF) Let p0; p1 be parent solutions.
- As previously suggested, the termination condition can be given by, for example, a fixed number of iterationsL in the outer loop.

### 4 Landscape analysis

- The link between landscape and search algorithm is given by the NS operators used in the algo- rithm.
- For the same PFSP but with simpler NS operators, similar experiments reported in Reeves [6] found such a landscape did occur.
- As discussed in [4, 2, 6], the existence of a big valley structure can be examined by first generating a set of random local optima and then observing the correlation be- tween their objective function values and their distances to the nearest global optimum, and/or their average distances to other local optima.
- Extensive preliminary experiments found only two distinct global optima for the ta011 problem, very close to each other in terms of the precedence-based distance (the distance is two) and only one global optimum for ta021 problem; however one cannot rule out the possibility of finding other different global optima by continuing the search.
- Therefore, the use of the easily-computed precedence-based distance appears to be justified, and the ‘big valley’ structure can be assumed to hold for this neighbourhood.

### 5 MSXF-GA for PFSP

- MSXF-GA provides a framework for traversing local op- tima without being trapped, by concentrating its attention on the area between the parent solutions and thus eventually finding a very good solution under the assumption of a ‘big valley’.
- MSXF-GA was applied to PFSP using the representative neighbourhood described in Section 2 and the precedence-based distance.
- Do 1. Select two schedulesp0; p1 probabilistically from the population with a probability inversely proportional to their ranks.
- If q’s makespan is less than the worst in the population, and no member of the current population has the same makespan asq, replace the worst individual withq.

### 6 Experimental results

- In Section 4, the existence of a big valley structure became clear for the relatively small-size PFSP instances.
- An adaptive multi-start method (AMS) in which new local search Parent0 Parent1 Offspring MSXF Figure 4: Navigated local search by MSXF-GA: A new search is started from one of the parents and while no other good solutions are found, the search ‘navigates’ towards the other parent.
- Table 1 summarizes the performance statistics of MSXF-GA for a subset of Taillard’s benchmark problems together with the results found by Nowicki and Smutnicki using their tabu search implementation[5] and the lower and upper bounds, taken from the OR-library [1].
- In all, 30 runs were completed for each problem under the same conditions but with different random number seeds.
- The results for larger problems are not as impressive as those of50 20 problems, but still good enough to support their hypothesis.

### 7 Conclusions

- The landscape for the Permutation Flowshop Scheduling Problem with stochastic local search and the representative neighbourhood structure has been investigated.
- The experimental analysis using20 10 and20 20 Taillard benchmark problems shows the existence of a ‘big valley’ structure for PFSP.
- This suggests a well-designed AMS method, such as MSXF-GA in which new local search is concentrated in a region between previously found local optima should be effective in finding near-optimal solutions.
- MSXF-GA for the PFSP is implemented using the neighbourhood operator and applied to more challenging benchmark problems.
- Experimental results demonstrates the effectiveness of the proposed method.

Did you find this useful? Give us your feedback

##### Citations

[...]

6,373 citations

### Cites background from "Permutation flowshop scheduling by ..."

...…Design – Consiglio and Zenios (1999) • Neural Network Training – Kelly, Rangaswamy and Xu (1996) • Job Shop Scheduling – Yamada and Nakano (1996) • Flow Shop Scheduling – Yamada and Reeves (1997) • Graph Drawing – Laguna and Marti (1999) • Linear Ordering – Laguna, Marti and Campos (1997) •…...

[...]

801 citations

711 citations

### Cites background from "Permutation flowshop scheduling by ..."

...…– Consiglio and Zenios (1996) Neural Network Training – Kelly, Rangaswamy and Xu (1996) Job Shop Scheduling – Yamada and Nakano (1996) Flow Shop Scheduling – Yamada and Reeves (1997) Graph Drawing – Laguna and Marti (1997) Linear Ordering – Laguna, Marti and Campos (1997) Unconstrained…...

[...]

273 citations

264 citations

### Cites methods from "Permutation flowshop scheduling by ..."

...The only exceptions are Reeves [75], Yamada and Reeves [107], and Ross and Tuson [103], who presented results on standard benchmark problems taken from Tailard [112]....

[...]

...NRX is, in fact, a neighborhood search algorithm, as the MSFX operator proposed by Yamada and Reeves [107]....

[...]

...Among them, Reevest al....

[...]

...only exceptions are Reeves [75], Yamada and Reeves [107], and Ross and Tuson [103], who presented results on standard benchmark problems taken from Tailard [112]....

[...]

##### References

[...]

6,373 citations

2,173 citations

2,164 citations

### "Permutation flowshop scheduling by ..." refers methods in this paper

...1 describes the outline of the MSXF-GA routine for the PFSP using the steady state model proposed in [8, 11]....

[...]

1,939 citations

### "Permutation flowshop scheduling by ..." refers methods in this paper

...Table 1 summarizes the performance statistics of MSXF-GA for a subset of Taillard’s benchmark problems together with the results found by Nowicki and Smutnicki using their tabu search implementation[5] and the lower and upper bounds, taken from the OR-library [1]....

[...]

...It can be seen that the results for50 20 problems are remarkable: the solution qualities of our best results are improved over those found in [5] for most of the problems, and some results (marked in bold letters) are even better than the existing best results reported in the OR-library....

[...]

1,314 citations

### "Permutation flowshop scheduling by ..." refers methods in this paper

...1 describes the outline of the MSXF-GA routine for the PFSP using the steady state model proposed in [8, 11]....

[...]