Job Shop Scheduling using CQM

Job shop scheduling is an optimization problem where the goal is to schedule jobs on a certain number of machines according to a process order for each job. The objective is to minimize the length of schedule also called make-span, or completion time of the last task of all jobs.

This example demonstrates a means of formulating and optimizing job shop scheduling (JSS) using a constrained quadratic model (CQM) that can be solved using a Leap hybrid CQM solver. Contained in this example is the code for running the job shop scheduler as well as a user interface built with Dash.

Usage

To run the job shop demo with the user interface, from the command line enter

python app.py

This will launch a local instance of the application on localhost. The default run location is http://127.0.0.1:8050/. Open the location in a web browser to view the application.

To run the stand-alone job shop demo (without the user interace), use the command:

python job_shop_scheduler.py [-h] [-i INSTANCE] [-tl TIME_LIMIT] [-os OUTPUT_SOLUTION] [-op OUTPUT_PLOT] [-m] [-v] [-q] [-p PROFILE] [-mm MAX_MAKESPAN]

This will call the job shop scheduling algorithm for the input instance file. Command line arguments are defined as:

• -h (or --help): show this help message and exit
• -i (--instance): path to the input instance file; (default: input/instance5_5.txt)
• -tl (--time_limit) time limit in seconds (default: None)
• -os (--output_solution): path to the output solution file (default: output/solution.txt)
• -op (--output_plot): path to the output plot file (default: output/schedule.png)
• -m (--use_mip_solver): Whether to use the MIP solver instead of the CQM solver (default: False)
• -v (--verbose): Whether to print verbose output (default: True)
• -q (--allow_quad): Whether to allow quadratic constraints (default: False)
• -p (--profile): The profile variable to pass to the Sampler. Defaults to None. (default: None)
• -mm (--max_makespan): Upperbound on how long the schedule can be; leave empty to auto-calculate an appropriate value. (default: None)

There are several instances pre-populated under input folder. Some of the instances were randomly generated using utils/jss_generator.py as discussed under the Problem generator section.

Other instances were pulled from E. Taillard's list of benchmarking instances. If the string "taillard" is contained in the filename, the model will expect the input file to match the format used by Taillard. Otherwise, the following format is expected:

#Num of jobs: 5 
#Num of machines: 5 
                task 0            task 1            task 2            task 3            task 4      
  job id    machine    dur    machine    dur    machine    dur    machine    dur    machine    dur
--------  ---------  -----  ---------  -----  ---------  -----  ---------  -----  ---------  -----
       0          1      3          3      4          4      4          0      5          2      1
       1          3      4          2      0          1      0          0      1          4      0
       2          1      5          4      5          2      4          0      0          3      3
       3          4      1          3      4          0      2          2      0          1      2
       4          1      3          3      3          0      0          2      0          4      1

Note that:

• tasks must be executed sequentially;
• dur refers to the processing duration of a task;
• this demo solves a variant of job-shop-scheduling problem

The program produces a solution schedule like this:

#Number of jobs: 5
#Number of machines: 5
#Completion time: 22.0

                  machine 0               machine 1               machine 2               machine 3               machine 4       
  job id    task    start    dur    task    start    dur    task    start    dur    task    start    dur    task    start    dur
--------  ------  -------  -----  ------  -------  -----  ------  -------  -----  ------  -------  -----  ------  -------  -----
       0       3       16      5       0        5      3       4       21      1       1        8      4       2       12      4
       1       3        6      1       2        5      0       1        4      0       0        0      4       4        9      0
       2       3       17      0       0        0      5       2       13      4       4       17      3       1        6      5
       3       2        9      2       4       19      2       3       14      0       1        4      4       0        1      1
       4       2       18      0       0        9      3       3       21      0       1       14      3       4       21      1

The following graphic is an illustration of this solution.

Generating Problem Instances

utils/jss_generator.py can be used to generate random problem instances. For example, a 5 * 6 instance with a maximum duration of 8 hours can be generated with:

python utils/jss_generator.py 5 6 8 -path < folder location to store generated instance file >

To see a full description of the options, type:

python utils/jss_generator.py -h

Model and Code Overview

Problem Parameters

These are the parameters of the problem:

• n : is the number of jobs
• m : is the number of machines
• J : is the set of jobs ({0,1,2,...,n})
• M : is the set of machines ({0,1,2,...,m})
• T : is the set of tasks ({0,1,2,...,m}) that has same dimension as M.
• M_(j,t): is the machine that processes task t of job j
• T_(j,i) : is the task that is processed by machine i for job j
• D_(j,t): is the processing duration that task t needs for job j
• V: maximum possible make-span

Variables

• w is a positive integer variable that defines the completion time (make-span) of the JSS
• x_(j_i) are positive integer variables used to model start of each job j on machine i
• y_(j_k,i) are binaries which define if job k precedes job j on machine i

Objective

Our objective is to minimize the make-span (w) of the given JSS problem.

Constraints

Precedence Constraint

Our first constraint, equation 1, enforces the precedence constraint. This ensures that all tasks of a job are executed in the given order.

(1)

This constraint ensures that a task for a given job, j, on a machine, M_(j,t), starts when the previous task is finished. As an example, for consecutive tasks 4 and 5 of job 3 that run on machine 6 and 1, respectively, assuming that task 4 takes 12 hours to finish, we add this constraint: x_3_6 >= x_3_1 + 12

No-Overlap Constraints

Our second constraint, equation 2, ensures that multiple jobs don't use any machine at the same time. (2)

Usually this constraint is modeled as two disjunctive linear constraints (Ku et al. 2016 and Manne et al. 1960); however, it is more efficient to model this as a single quadratic inequality constraint. In addition, using this quadratic equation eliminates the need for using the so called Big M value to activate or relax constraint (https://en.wikipedia.org/wiki/Big_M_method).

The proposed quadratic equation fulfills the same behaviour as the linear constraints:

There are two cases:

• if y_j,k,i = 0 job j is processed after job k:
  
• if y_j,k,i = 1 job k is processed after job j:
  
  Since these equations are applied to every pair of jobs, they guarantee that the jobs don't overlap on a machine. If -allow_quad is set to False, this mixed integer formulation of this constraint will be used.

Make-Span Constraint

In this demonstration, the maximum makespan can be defined by the user or it will be determined using a greedy heuristic. Placing an upper bound on the makespan improves the performance of the D-Wave sampler; however, if the upper bound is too low then the sampler may fail to find a feasible solution.

References

A. S. Manne, On the job-shop scheduling problem, Operations Research , 1960, Pages 219-223.

Wen-Yang Ku, J. Christopher Beck, Mixed Integer Programming models for job shop scheduling: A computational analysis, Computers & Operations Research, Volume 73, 2016, Pages 165-173.

License

Released under the Apache License 2.0. See LICENSE file.