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An improved ant colony algorithm for the Job Shop

Scheduling Problem Guillermo Leguizamón Martin Schütz

Presented by Meredith Taylor

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What is the Job Shop Scheduling Problem (JSSP)?

• NP-Hard

•Finite set Τ of n jobs to be processed on a finite set Μ of m machines .

•Each job Τi must be split between the m machines and consists of a chain of m operations

•Each job’s operations must be scheduled in a predetermined given order (precedence constraint)

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What is the Job Shop Scheduling Problem (JSSP)?

• N = hjfdh operations in total where oik is the operation of job Ti which has to be processed on machine Mk for an uninterrupted processing time pik.

•Each machine can process only one job and each job can be processed by only one machine at a time (capacity constraint).

•All the jobs are completely independent of each other

What is the Job Shop Scheduling Problem (JSSP)?

•The duration in which all operations for all jobs are completed is referred to as the makespan Cmax

•The objective is to determine the starting times (tik ≥ 0) for each operation in order to minimise the makespan while satisfying all constraints:

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What is the Job Shop Scheduling Problem (JSSP)?

• Different types of Schedules

What is the Job Shop Scheduling Problem (JSSP)?

•What Schedules are optimal?

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Real World applications of JSSP

• Operations management

•Manufacturing and assembly situations

•Need to take into account: manpower, number of machines available , cost of operating machines, outside suppliers

•A particular process anticipated by a manufacturer will consist of a number of jobs which must be carried out in a certain order for successful completion.

•Scheduler or project manager must optimise the schedule when planning a production process.

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Previous approaches to solving JSSP

• Genetic algorithm • Described in C. Bierwirth and D. Matffeld. Production scheduling and rescheduling with genetic algorithms. Evolutionary Computation, 7(1):1–17, 1999.

•The schedule builder in Figure 3 generates active and non-delay schedules from a permutation of operations. The parameter δ (0 ≤ δ ≤ 1) controls the type of schedule to be generated.

•Non-delay schedules are obtained by setting δ = 0, all active schedules can be generated with δ = 1

• Simulated Annealing

• Threshold Accepting

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Ant-based system for the JSSP

•An instance of the JSSP represented as a graph.

•Each node represents an operation

•Nodes are connected by 2 kinds of edges

•Directed edges represent the precedence between operations for the same job

•Undirected edges represent possible paths to follow by the ants if the problem constraints are satisfied

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Ant-based system for the JSSP

n = 3 jobs m = 2 machines

Node 1 represents the 1st operation of job 1, node 2 represents the 2nd operation of job 1, etc.

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Ant-based system for the JSSP

n = 3 jobs m = 2 machines

Possible solution: p = (0,1,4,5,2,3,6,7)

In general: node i represents the (i mod m+1)th operation of job (i div(m+1))+1

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Ant-based system for the JSSP

•Each ant builds a solution “walking” from node 0 to node 7 by visiting all nodes, i.e. scheduling each operation to complete the schedule

•Decision about the path will depend on amount of pheromone laid on the connections and the corresponding heuristic values.

•Similar to traveling salesman problem with precedence rules and designated start and end nodes

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Ant-based system for the JSSP

The variables denoting the intensity of pheromone on connection at time are defined as:

Where is a coefficient representing pheromone evaporation.

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(1)

Ant-based system for the JSSP

where a is the actual number of ants in the colony and is the quantity per unit of length of trail substance (pheromone for real ants) laid on connection (i,j) by the e-th ant at time t and is given by

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(2)

Ant-based system for the JSSP

Where Le is the makespan found by the eth ant and Q is a constant

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(2)

Ant-based system for the JSSP

While building a permutation of operations, the probability that ant e schedules operation j provided that i was the last operation scheduled by ant e is given by:

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(3)

Ant-based system for the JSSP

Where Se(t) is the set of schedulable operations still not scheduled by ant e at time t, and ηij(t) represents the visibility or local heuristic.

α and β control the relative importance of pheromone versus visibility (local heuristic).

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(3)

Ant-based system for the JSSP

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Ant-based system for the JSSP Their new improved Ant System for the Job Shop Scheduling Problem (AS-JSS):

AS-JSS-δ •Aims at processing the solutions or permutation found in each cycle

•Similar to AS-JSS) except that between lines 9 and 10, the Hybrid Scheduler Builder (HSB) described in Figure 3 is integrated

•aims at the improvement of the solutions found so far.

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Ant-based system for the JSSP

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Ant-based system for the JSSP Their new improved Ant System for the Job Shop Scheduling Problem (AS-JSS):

AS-JSS-δ •Aims at processing the solutions or permutation found in each cycle

•Similar to AS-JSS except that between lines 9 and 10,the Hybrid Scheduler Builder (HSB) described in Figure 3 is integrated

•Aims at the improvement of the solutions found so far.

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AS-JSS-δ

•AS-JSS-δ receives a permutation obtained for a particular ant and then builds a new one by applying the steps described in the HSB

•The application of HSB is biased according to parameter δ.

•any improvement on some solution could influence the pheromone laid on the paths involved in the solution.

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Experiments and Results •The performance of AS-JSS and AS-JSS- δ applied to several instances of the Job Shop Scheduling problem was studied.

• Instances Taken from OR Library http://people.brunel .ac.uk/~mastjjb/jeb/info.html

•Parameters for AS-JSS: α = 1, β = 5, a = 30, ρ = 0.5

•Parameters for AS-JSS-δ: α = 1, β = 1, a = 30, ρ = 0.5 and δ ∈ {0,0.1,0.3,0.5,0.7,0.9,1}

•Number of cycles = 3000

•Both algorithms were run 10 times using different random seeds for each instance considered.

•Q was set to the best known value of the respective instance.

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http://people.brunel.ac.uk/~mastjjb/jeb/info.html http://people.brunel.ac.uk/~mastjjb/jeb/info.html http://people.brunel.ac.uk/~mastjjb/jeb/info.html http://people.brunel.ac.uk/~mastjjb/jeb/info.html http://people.brunel.ac.uk/~mastjjb/jeb/info.html http://people.brunel.ac.uk/~mastjjb/jeb/info.html http://people.brunel.ac.uk/~mastjjb/jeb/info.html http://people.brunel.ac.uk/~mastjjb/jeb/info.html http://people.brunel.ac.uk/~mastjjb/jeb/info.html

Influence of δ

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Comparison between AS-JSS & AS-JSS-δ

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Evaluation of approach

• Approach to guide the ant system (AS-JSS-δ) to specific areas of the search space in order to look for the optimal schedule.

•No results obtained that were better than best known result

•In all test cases (at least ones reported in paper) AS-JSS-δ produced either better or exactly the same results as AS-JSS

∀β is was different for each algorithm.

•For AS-JSS β = 5

•For AS-JSS-δ β = 1

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Evaluation of approach

•Proposes in the future considering the design of an Ant System which generates either, non- delay or active schedules controlled by δ but during the construction phase.

•Could avoid extra computation time and perhaps lead to an extra improvement in the quality of the results.

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Thank You

Questions?

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