kartik96/animated-archer

Running

All tests

ALL_test_16t6m();
ALL_test_17t5m();

TS

source code file: TS.m
All test files match TS_*

example

iterations = 50;
m = 3;
n = 6;
J = [2, 3, 4, 6, 2, 2];
[costs, bestSol] = TS(J, m, n, 10, iterations, @cost, @getBestNeighbor);

SA

source code file: SA.m
All test files match SA_*

example

iterations = 50;
m = 3;
n = 6;
S = initSolution(m,n);
J = [2, 3, 4, 6, 2, 2];
sT = findStartTemp(J, m);
alpha = 0.85;
iterationsAtTemp = 2;
fT = sT*(alpha^(iterations/iterationsAtTemp));
[costs, bestSol] = SA(S, J, m, n, iterationsAtTemp, sT, fT, alpha, @cost, @gen_neighbor);

GA

source code file: GA.m
All test files match GA_*

example

iterations = 50;
m = 3;
n = 6;
J = [2, 3, 4, 6, 2, 2];
[costs, bestSol] = GA(J, m, iterations);

PSO

source code file: PSO.m PSO_lbest.m (better performance)
All test files match PSO_*

example

iterations = 50;
m = 3;
n = 6;
J = [2, 3, 4, 6, 2, 2];
[costs, bestSol] = PSO_lbest(J, m, n, 1500, iterations, @cost);

ACO

source code file: ACO.m
All test files match ACO_*

example

iterations = 50;
m = 3;
n = 6;
J = [2, 3, 4, 6, 2, 2];
ants = 5;
[costs, bestSol] = ACO(J, m, n, ants, iterations, 0.2, @cost);

Problem Description

Job shop scheduling is an optimization problem in which n jobs J1, J2, …, Jn of varying sizes are given. These jobs need to be scheduled on m identical machines, while trying to minimize the makespan. The makespan is the total length of the schedule (that is, when all the jobs have finished processing).

Problem Formulation

Variables

n, number of Jobs
m, number of identical Machine
J, an array of each Jobs’ weight
S, an array of each Jobs’ Schedule

Constraints

m > 1
n > m
∀ s ∈ S, 1 <= s <= m

Cost Function

Time takes the longest scheduled machine to finish. See cost.m

Goals

Minimize cost function
Minimize number of iterations for each algorithm
Find the best algorithm for the problem

Simple Example

J = (2,3,4,6,2,2)
S = (1,2,2,3,1,1)
Cost = 7
This setup is optimal

Tabu Search

Neighbors of each schedule

Each schedule will have (m-1)*n neighbors, where m is the number of machines, and n is the number of jobs. Neighbors will only have one job scheduled on a different machine.

Finding the best neighbor

In order the find the neighbor with the lowest cost, the algorithm will loop through every valid neighbor and evaluate its cost. The neighbor with the lowest cost will be selected as the best neighbor.

Tabulist

The list length of the tabulist is user-defined.
The tabulist acts like a queque (first in first out)
The oldest move will be deleted when a new move is appended.
A new move is appended every time after finding a best neighbor.

Simulated Annealing

Neighbors of each schedule

Each schedule will have (m-1)*n neighbors, where m is the number of machines, and n is the number of jobs. Neighbors will only have one job scheduled on a different machine.

Finding a suitable start temperature

Assume the max change is the MAX of
- total time of all job divide by number of machines.
- max time of a single job.
Formula to find the max
- Temp_start = -1 * max_change / ln(p_0), where p_0 is 0.85
Start temperature is not calculated within SA, need to be calculated before execute the SA.
- see “findStartTemp” in “SA_test.m”.

Cooling Schedule

Using geometric cooling schedule.
Final temperature should close to zero but not equal to zero.
alpha = 0.75 ~ 0.9 is commonly used.
iteration
- a constant.
- number of iteration for each temperature.

Genetic Algorithm

Overview

This part uses Genetic Algorithm to find the optimal solution for the job scheduling problem. The process was inspired by the evolution of organisms in natural. It employs random crossover, mutation and evolution to achieve the goal of finding the optimal scheduling for a set of given jobs. This process is based on the stock Genetic Algorithm given by the professor.

Initial state

The population size is set to 100
Chromosome length depends on the range of the possible output
Crossover Probability was set to 95%
Mutation probability was set to 5%
There will be 2 sites of mutation, when the mutation event occurs

Crossover

The crossover will exchange chromosome information at a specified crossover site, which is generated randomly.
After each crossover, evolve will be called, and the fittest of the older population, or its offspring will survive.

Evolve

The evolve function will maximize the model function, 1/(1+cost), which is the same as to minimize the cost
The old and the new population will be compared, and the fitter of the two will get passed to the next generation

Mutate

A given number mutation sites were generated, and the binary bits at the generated mutation sites will be flipped
Evolve function will be called, and the older generation and the newer generation will be compared, the fittest of the two will get passed on to the next generation

Particle Swarm Optimization

Overview

This part uses the Ring Topology or lbest Particle Swarm Algorithm to find optimal solution for job scheduling problem. Each particle is communicating with four of its adjecent neighour. In each iteration, each particle calculates its speed based on the best solution in its neighbour and its personal best. Speed and location is defined in n dimensions.

Initial state

All particals starts with 0 speed at all n directions.
All particals starts at location randomly assigned between 1 ~ m in all dimensions.
Local best solution is the same as partical’s location
Neighbor best solution in each particle is the best solution in four of its neighours based on neighbor index.

Local search criteria

Speed is calculated based on each particle’s personal best solution and the best solution of its neighbor. c1 = 1.4944, c2 = 1.4944, w = 0.9, v_t+1ⁱ = w× v_tⁱ+c₁r₁ⁱ(pbest_tⁱ-x_tⁱ)+ c₂r₂ⁱ(Nbest_tⁱ-x_tⁱ)
The new solution is calculated by adding its previous location and its new speed, x_t+1ⁱ = x_tⁱ+v_t+1ⁱ
When the new cost of the new location is smaller than a particle’s local best, it updates its local best and update its neighbour’s neibour best when applicable.
Asynchronous update method is used to reduce run time load requirement, neighbor best is updated when all partical finishes its calculation for its current round.

Termination Criteria

The algorithm is terminated when set number of particals completes set number of iterations.
The number of particals determines the amount of exploration and the amount of iterations determines the amount of exploitation.

Ant Colony Optimization

Overview

This part uses Ant Colony System to find the optimal solution for the job scheduling problem. The process is similar to find a shortest path between two nodes on an weighted tree graph.

Initial state

All ants starts at layer 0 of the tree, which means no job has been scheduled.
All routes has initial pheromone of 1.
pheromone will decrease 40% after each round.

Local search criteria

Local search depends on the number of pheromone, and the cost to move the next level.
The cost is calculate by the the extra number of time required for including the next job in certain machine. The cost can be zero.
Using experience vs Explore the new scheduling
1. a rand value is generate to compare with r_0
2. if the rand value is smaller than r_0, the local search will select the route with max amount of pheromone
3. otherwise, it will do a roulette wheel selection based on ( pheromone / (route-cost + 1))

Pheromone deposit

only the best ants in each round can deposit pheromone on its path.
the number of pheromone deposited equals to ( 1 / best-ant-total-cost).

CPU Time Usage Data

Total time for 1000 iterations

data set	GA	PSO	TS	SA	ACO
16t6m	23.15	6.490	6.677	0.062	49.23
17t5m	22.62	6.209	5.600	0.047	44.15

Time of Convergence

data set	GA	PSO	TS	SA	ACO
16t6m	16.97	2.726	0.133	0.012	0.985
17t5m	9.048	5.588	0.168	0.011	0.883

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Octave/Matlab Implementation for Job Shop Scheduling Problem

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