The subject of this article is the job shop
scheduling problem and methods for solving this
problem. It contains model for the task of schedule
optimizing for partially ordered jobs on machines
with different productivity in the presence of idle time
and algorithm for getting initial permissible solution

Ilchuk, O.

Keller, R.

SocioEconomic Challenges Journal (Sumy State University)

  
 
MINISTRY OF EDUCATION AND SCIENCE OF UKRAINE 
SUMY STATE UNIVERSITY 
UKRAINIAN FEDERATION OF INFORMATICS 
 
 
 
 
 
 
 
 
PROCEEDINGS  
OF THE V INTERNATIONAL SCIENTIFIC 
CONFERENCE 
ADVANCED INFORMATION 
SYSTEMS AND TECHNOLOGIES 
 
AIST-2017 
(Sumy, May 17–19, 2017) 
 
 
 
 
 
 
 
SUMY 
SUMY STATE UNIVERSITY 
2017 
The Vth International Conference «Advanced Information Systems and Technologies, AIST 2017» 
17-19 May 2017, Sumy, Ukraine 
83 
 
The task of schedule optimizing for partially ordered 
jobs on machines with different productivity in the 
presence of idle time 
Raymond Keller, Olesia Ilchuk 
 
National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", raybed@ukr.net 
 
Abstract – The subject of this article is the job shop 
scheduling problem and methods for solving this 
problem. It contains model for the task of schedule 
optimizing for partially ordered jobs on machines 
with different productivity in the presence of idle time 
and algorithm for getting initial permissible solution. 
Keywords – job shop scheduling; optimizing; initial 
acceptable solution; algorithm; theory of schedule. 
I. INTRODUCTION 
Scheduling theory deals with "ordering" and 
"scheduling". Ordering - it queuing operations performed 
by a machine, and scheduling - a task action sequences 
for several cars. In this scientific paper the "scheduling", 
namely the class of problems in which we have a service 
system that consists of machines. Each machine performs 
(some basic tasks). These operations are partially ordered, 
that perform one operation may depend on a previous 
execution did not depend. However, the duration of the 
operation is not a constant, it depends on the machine on 
which it runs. At the time axis may be "gaps" that forced 
the machine idle periods, when it cannot perform any 
operations. The set of machines, transactions and 
operations disciplines relevant purpose machines called 
maintenance. Scheduling this process means that for 
every transaction on the time axis corresponds to the 
subset in which the operation is performed in accordance 
machine. The main objective of this class of problems is 
to reduce maintenance time. 
II. EXISTING APPROACHES TO SOLVING PROBLEMS 
SCHEDULING 
A. Local Search 
Local Search - search carried out by local search 
algorithms, a group of algorithms which search is 
conducted only on the basis of the current state and 
earlier passed states are not counted and not memorable. 
The main purpose of the search is not an optimal path 
to the target point, and some optimization objective 
function, so the problem solved similar algorithms, called 
optimization problems. To describe the state space such 
problems using state space landscape, this presentation of 
the problem is to find the state of global maximum (or 
minimum) at this landscape. 
B. Branch and bound method 
Algorithmic method for finding the best solutions of 
various optimization problems, in particular, NP-
complete problems, such as the problem of traveling 
salesman. Method is a combinatorial (sorting algorithm) 
with the rejection of subsets of feasible solutions that do 
not contain optimal solutions. The idea of the method can 
be described by the example of finding a minimum or 
maximum of a function f (x) on the set of admissible 
values of x. The method comprises two procedures: 
branching and evaluation (search boundaries).The first is 
to partition the set of feasible solutions to smaller subsets 
(technology "divide and conquer"). The procedure can be 
repeated recursively for all subsets. The result will be 
obtained tree is called a tree branch and bound tree or 
search nodes which are received subset. The assessment 
procedure is to find upper and lower bounds for the 
optimal value on a subset of feasible solutions.Screenings 
decision is made in the following way. To minimize 
problems, if the lower limit for the subset A search tree 
than the upper limit of any of the previously discussed 
subsets of B, then A can be eliminated from further 
consideration. The smallest of the received top ratings 
memorize in the variable m. Any node tree search, the 
lower limit is greater than the value m, can be excluded 
from further consideration. If the lower boundary 
coincides with the node tree top, this value is the 
minimum of the function at the appropriate subset. 
C. Genetic algorithms 
Genetic Algorithms - adaptive search, which recently 
often used for solving functional optimization. They are 
based on the genetic processes of biological organisms, 
biological populations develop over several generations, 
obeying the laws of natural selection and the principle of 
"survival of the fittest". 
The task is coded so that its solution could be 
represented in the form similar to the array data structure 
of the chromosome. This array often called just that: 
"chromosome". Random array creates a certain amount of 
initial elements "Persons" or initial population. 
Individuals are valued using the tool accessories, which 
resulted in each person is assigned a certain value fitness, 
which determines the possibility of the survival of the 
individual. Then using obtained values adaptability 
selected persons admitted to crossing (selection). Persons 
The Vth International Conference «Advanced Information Systems and Technologies, AIST 2017» 
17-19 May 2017, Sumy, Ukraine 
84 
 
used "genetic operators" (in most cases an operator 
intersection (Crossover) and mutation operator 
(mutation)), thus creating the next generation of people. 
Those next generations also evaluated using genetic 
operators and performed selection and mutation. Yes, 
simulated evolutionary process continues several life 
cycles (generations) until stopping criterion satisfied 
algorithm 
Genetic algorithm consists of several stages: 
1. creation of initial population (chromosome create 
for each individual) 
2. the calculation of each individual fitness function 
(that it shows who fit best in this population) 
3. Selection of the best representatives for further 
education of offspring 
4. crossover 
5. mutation 
6. When (4) get the children, some of which passes 
through (5). The output obtain offspring 
7. Selection of parents and children in the next 
generation 
8. return to step (2) if the values that give our 
children is not satisfied 
D. Simulated annealing 
Simulated annealing - general algorithmic method of 
solving the problem of global optimization, especially in 
discrete and combinatorial optimization, in which the 
procedure for finding global solution simulates the 
physical process of annealing. 
Exotic names associated with the algorithm of 
simulation methods in statistical physics, based on the 
technique of Monte Carlo. Investigation of the crystal 
lattice and behavior of atoms slow cooling of the body 
led to the appearance of the light probabilistic algorithms 
that have proven extremely effective in combinatorial 
optimization. It was first observed in 1983. Today this 
algorithm is popular among practitioners because of its 
simplicity, flexibility and efficiency, as well as among 
theorists, since the algorithm cannot analytically 
investigate its properties and prove asymptotic 
convergence. 
Simulated annealing refers to a class of local search 
algorithms threshold. At each step of the algorithm for 
this decision in its neighborhood chosen a solution and if 
the difference in the objective function between the new 
and the current decision does not exceed the 
predetermined threshold, the new solution replaces the 
current one. Otherwise, select the new adjacent solutions. 
The general scheme of threshold algorithm can be 
represented as follows. 
E. Taboo search 
Taboo search is a meta-heuristic algorithm, local 
search is to prevent it from falling into the trap of 
premature local optimum by prohibiting the movement of 
those who are forced to return to previous decisions and 
cyclic operation. Taboo search begins with the initial 
decision. At each iteration margin generated solutions 
and best for this neighborhood gets a new solution. 
Certain attributes stored in previous decisions taboo-list, 
which is updated at the end of each iteration. Choosing 
the best solution going around so that it does not take any 
of the prohibited attributes. Most acceptable solution is 
now being updated when new and better solution to the 
current limit. The procedure continues until the fail any 
of the two criteria breakpoint, which is the maximum 
number of iterations performed and the maximum 
number of iterations during which the current solution is 
not improving. 
III. THE FEASIBILITY OF THE RESEARCH 
An interesting example is the guild task scheduling 
theory routing generalizing guild tasks commonly known 
metric traveling salesman problem. Performances such 
problems appeared independently in considering 
problems arising both in manufacturing and in the service 
industry. Most tasks scheduling theory is NP-difficult. 
Sorting algorithms exponential type require 
considerable computational cost even when solving 
examples of average dimensions. Therefore, one of the 
important areas of research is accelerating algorithms for 
NP-hardness. Currently, this area has gained a huge 
number of supporters among researchers engaged in 
computer mathematics. Here are the main reasons for the 
popularity: 
 there are many optimization problems that require 
solutions, and most NP-complex. For many of 
them not necessarily find the exact solution, but 
rather to build about, but in a short time, the more 
successfully cope with approximate algorithms; 
 in practice, most optimization problems are very 
complex, because it includes many additional 
restrictions that do not allow to find a satisfactory 
approximate solution. But usually approximate 
algorithms for simpler versions of the same tasks 
suggest new ideas for heuristics, which then 
successfully used for practical tasks; 
 in terms of exact solutions of NP-almost all 
challenges are equivalent to each other. 
Construction of approximate algorithms or proved 
the impossibility of constructing differences in the 
classes P and NP makes it possible to compare th 
NP-complex tasks by how they can be 
approximated. 
Thus, we see that the current algorithms can be improved 
at the expense of speeding. 
IV. FORMULATION OF THE PROBLEM 
The problem purpose partially ordered work to handle 
their various capacities on machines that contain certain 
periods of downtime if downtime to minimize the time of 
execution of works. Suppose we have a set of works 
The Vth International Conference «Advanced Information Systems and Technologies, AIST 2017» 
17-19 May 2017, Sumy, Ukraine 
85 
 
niJJ i ,1},{   і and set of cars 
mjMM j ,1},{  . Time for each job on each 
machine defined matrix 
mnijx )( . Also considered a 
given set period of inactivity klHH l ,1},{  , along 
with three vectors ),,( 1 k  , ),,( 1 krr  , 
),,( 1 k  , which determine the length of downtime, 
the number of machines to which he belongs, and the 
time it starts, respectively. The problem can be 
represented by two vectors — ),,( 1 nss  , 
),,( 1 nzz  ,  which specifies the number of machines 
on which work is scheduled and the start of its 
implementation. Based on the above notation, a formal 
model of the problem can be represented as follows: 
mjnixij ,1,,1,0)(  , 
kll ,1,0  , 
nimsi ,1},,,1{   , 
klmrl ,1},,,1{   , 
nizi ,1,0  , 
kll ,1,0  , 
For all },,1{},,,1{ nni     such that 
 zzss ii  , : 
zxz iisi      (1) 
For all },,1{},,,1{ kni     such that 
  ii zrs , : 
 iisi xz     (2) 
For all },,1{},,,1{ kni     such that 
ii zrs   , : 
iz       (3) 
min)(max
,1

 i
isi
ni
xz   (4) 
Limit (1) sets no crossing of the planned works on the 
same machine, constraints (2) and (3) - not crossing 
works and downtime. Limitation (4) - objective function 
that provides minimizing completion time of the most 
recent (v. BC. Guillotine cutting). 
V. ALGORITHM TO OBTAIN AN INITIAL ACCEPTABLE 
SOLUTION 
Step 1. Choose a car with the most power. 
Step 2. Go to Step 4. 
Step 3. CHOOSE car with less power. 
Step 4. CHOOSE task with the greatest of less than or 
equal intervals to stop the machine. 
Step 5. delete the selected task from the list available. 
Step 6. Set the selected task in the processing machine. 
Step 7. If the current machine is not the least power, then 
go to Step 3. 
Step 8 select the first car 
Step 9. Go to Step 11. 
Step 10. Step the Next machine 
Step 11. CHOOSE task with the greatest of less than or 
equal intervals to stop the machine. 
Step 12. If the problem is not selected TO CHOOSE next 
interval after stopping the car OTHERWISE go to step 
14. 
Step 13. Go to Step 11. 
Step 14. Set the time the machine flat end time of the end 
processing tasks. 
Step 15. If the current machine is not the last, then go to 
Step 10. 
Step 16. CHOOSE car with the lowest end of time. 
Step 17. delete the selected task for this machine from a 
list. 
Step 18. Set the selected task in handling this machine. 
Step 19. If the list of available tasks does not blank, then 
go to step 8. 
CONCLUSIONS 
As the previous review, all known methods for 
scheduling the task to be performed with NP-complexity. 
Given the current trends in computing, it should focus on 
reducing the complexity of existing algorithms, which 
will accelerate their work and productivity software that 
will use them. Were considered well-known methods for 
solving the above problem and algorithm to obtain an 
initial acceptable solution.  
REFERENCES: 
[1] Hulyanytskyy L, Turin V. “Mathematical model of a class of 
independent tasks scheduling machines”, Computer mathematics. 
Kyiv Institute of Cybernetics, vol. 2, 2012, pp. 1-6. 
[2] S. Kolliopoulos, G. Steiner. “Approximation algorithms for 
minimitation the total weighted tardiness on single machine”, 
2006, pp.1-10  
[3] A. Lazarev, E Rashidovich “Theory of schedules”, Moscow, 2011, 
pp. 222. 


The task of schedule optimizing for partially ordered jobs on machines with different productivity in the presence of idle time

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