An algorithm for a Parallel Machine Problem with Eligibility and Release and Delivery times, considering setup times Manuel Mateo Management Department, Universitat Politècnica Catalunya. Barcelona (Spain) HAROSA, Barcelona (10/07/13)

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times 2 Summary Introduction: the times, the eligibility and the setup times Notation and definition of the problem Pm/r j,q j,s j,M j /c max Proposed algorithms Initial Solution Heuristic algorithm Genetic Algorithm (crossover, mutation, local search) Computational experiments and results Conclusions Remark: this work is based on the Master Thesis in Engineering done by David Miquel.

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Introduction: the times 3 The manufacturing of products is usually divided in operations or phases of transformation. Usually one of them becomes the bottleneck of the process. In the presented problem, we suppose this bottleneck is an intermediate phase. Therefore, some operations are done before (the total time to work them out leads to a release time) and some others are done after (their total time is called delivery or queue time). jj1j2j3j4j5 p 1,j p 2,j p 3,j bottleneck j rjrj pjpj qjqj

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Line 4 Line 3 Line 2 Line 1 Introduction: eligibility Manufacturing plants usually have several machines or assembly lines (i.e., parallel machine). There are several products to be manufactured. A usual situation is a product that is assigned to a machine (line) and will be only manufactured in that machine. 4 ABC INITIAL SITUATION D

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Low-level machines Medium-level machines High-level machines Introduction: eligibility (II) But more product-machine assignments are possible: 5 Line 1 Line 2 Line 4 1 lDC Line 3 CONSIDERED SITUATION BDC BDC BDC 2 l A Medium-level Jobs Machines High-level Medium-level Low-level A B D C High-level Low-level

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times There are no setup times if the jobs classified in one level are done in the machines of the same level: If the jobs in one level are assigned in the machines of different level, setup times appear (between the schedule of jobs belonging to different levels): Introduction: the setup times 6 Medium-level Jobs Machines High-level Medium-level Low-level High-level Low-level Medium-level Jobs Machines High-level Medium-level Low-level High-level Low-level

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times 7 Notation of the problem The machines are distributed among p groups or levels (k=1,…,p). Particularly, we propose an algorithm for p=3 (high-level, medium-level and low-level). A set of n jobs (j=1,…,n) to be scheduled on m parallel machines (i=1,…,m). Given a job j, it is known: the processing time p j for the operation, the release time r j (also called head times), the delivery or queue time q j (also tail times), the associated level l j. Any machine i and job j is classified into one of the levels.

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Notation of the problem (II) The n jobs to be manufactured are also divided in three groups: J i is the subset of jobs of level i, with |J i |=n i. n=n 1 +n 2 +n 3 Eligibility restrictions: A machine in the level k can manufacture jobs of its own level k and also of levels k+1,…,p. The processing time of a job is the same for any machine. 8 Medium, k=2 Jobs Machines High-level (k=1) Medium-level (k=2) Low-level (k=3) High, k=1 Low, k=3

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times The problem Pm | r j, q j,s j,M j | c max The number of machines and jobs is initially known. A schedule is feasible if the next conditions are accomplished: Each machine processes at most one job at a time. A job is only processed in a single machine. Pre-emption is not allowed. Starting time is not lower than the release time: A job of level k is processed in a machine of level k or higher. Setup times are required when a job classified in a level is going to be manufactured after another of a different level. Machines are initially prepared for the jobs of their own level. It is not necessary another setup at the end of the last job scheduled if it is from a different level. 9

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times The problem Pm | r j, q j,s j,M j | c max (II) The processing in all machines for each job is the same, i.e., we consider identical parallel machines. Given t j the starting time for any job j, the completion time of the job is obtained: The makespan can be determined: c max = max{c j } For any feasible schedule, the objective is: Min {c max } 10

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Proposed algorithms 11 INITIAL SOLUTION HEURISTIC Preprocessing Insertion improvement Flexible improvement Genetic Algorithm (GA) METAHEURISTIC

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times At each level (3 times), it is necessary to solve a problem of a single machine (m k =1) or parallel machines (m k >1). Gharbi & Houari (2002 ) Medium-level Jobs Machines High-level Medium-level Low-level High-level Low-level Initial solution (no setup times)

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Description of the Heuristic Sequences of jobs: SPT (shortest processing time) SRT (shortest release time) SQT (shortest queue time) LPT (longest processing time) LRT (longest release time) LQT (longest queue time) 6 random sequences 13 Determine the sequence of jobs j=1 For each position pos of j in the sequence, cmax(pos) Assign the job in pos such that cmax is minimum; j=j+1 j=n? SEQUENCE YES NO j=j+1

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Preprocess of the GA Initial Solution Insertion improvement Flexible improvement Genetic Algorithm

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times General view of the Genetic Algorithm Initial population: 1 (Initial solution + insertion + flexible improvements). 6 (SPT, SRT,SQT, LPT, LRT, LQT) 73 random sequences (generate 100 sequences + apply the heuristic) Selection: 10% of individuals with best fitness 90% roulette rule 15 Initial population Fitness Selection Crossover Mutation Local search (IMI) Regeneration end? SOLUTION YES NO Fitness

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Genetic Algorithm: crossover 16 Crossover Share many of the characteristics in the parents’ solutions P c = 50% Reference: Vallada & Ruiz (2011)

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Genetic Algorithm: mutation & local search 17 Mutation Shift P m = 50% (Vallada & Ruiz, 2011) Local search (IMI, Inter-Machine Insertion neighborhood) Two machines P ls = 100% (Vallada & Ruiz, 2011) Acceptance criterion: Makespans of both machines are reduced. Makespan of one machine is reduced, although the other is increased.

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Computational experience To check the efficiency of the algorithm, a set of instances similar to those used by Gharbi & Haouari (2002) are created:  # jobs (n = {20,50,100,200})  1000 instances  Jobs of high level, 10-30% of the total number; jobs of medium level, 10-50% of the total; jobs of low level, the rest.  # machines (m = {4,5,6,8,10})  Processing time: discrete uniform distribution.  Release and delivery times: discrete distribution with K={3,5}  Setup times: 3 sets with uniform distributions of [1, 3], [1, 9] and [1, 19].

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Computational experience (II) Initial study on 220 instances to determine the computing time: balance between the results and the computational cost. CPU time = n · (m/2) · (time) ms  Improvement  5%  time = 240 Complete study on 4000 instances with different configurations of parallel machines to determine the improvement on the c max of the initial solution. Time 30  6060    300 n = n = n = n = Total69 / / / / 220

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Some results 20 Number of jobs: Proportion of machines: nIHIH I GA 20 27,7%28,6% 50 32,4%33,2% ,1%34,5% ,6%34,0% CaseIHIH I GA 1m h > m m ; m h > m l 39,9%40,0% 2m m > m h ; m m > m l 38,1%38,4% 3m l > m h ; m l > m m 18,9%20,0% 4m h < m m ; m h < m l 20,7%21,6% 5m m < m h ; m m < m l 30,4%30,7% 6m l < m h ; m l < m m 48,6%48,7% 7m h = m m = m l 31,2%31,5%

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Window for the setup times: Influence of the parameter K ={3,5}. Some results (II) 21 Setup timeIHIH I GA 133,7%34,1% 233,4%33,8% 333,1%33,6% n K = 3K = 5 IHIH I GA IHIH 2032,8%34,3%23,0%23,4% 5039,4%40,8%26,2%26,3% 10042,3%43,1%26,6%26,8% 20038,7%39,1%24,9%25,1% Average39,6%40,2%25,5%25,6%

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Conclusions from the computational experience We have not shown the results, but the introduction of the heuristic on the initial population improves the quality of solutions. Once it is introduced:  Some proportions of machines can lead to a improvements of nearly 50%; the best ones are: m h > m m ; m h > m l m l < m h ; m l < m m On the other hand, improvements about 20% are obtained by: m b > m h ; m l > m m m h < m m ; m h < m l  The lower the parameter K is, the greater the improvement is (40% for K=3 and 25% for K=5).  Any number of jobs and any window for setup times give similar results (improvements around 30%).

An algorithm for Parallel Machine Scheduling Problem with Eligibility, Ready Dates and Delivery Times, considering setup times Conclusions We studied the problem of parallel machines with eligibility and release and queue times. The problem has the objective to minimize the makespan. The setup times are introduced when a job of a different level from the one of the previous one is going to be manufactured. We proposed a heuristic procedure, based on some initial sequences of jobs. A Genetic Algorithm is developed, which takes the advantages of the heuristic in the preprocess. Improvement respect to initial solution varies from 20% to 50%. About the current and future research:  We hope to tune the parameters of the Genetic Algorithms.  Another metaheuristic (for instance ILS) could be developed.

An algorithm for a Parallel Machine Problem with Eligibility and Release and Delivery times, considering setup times