Heuristic Algorithm for Uncertain Permutation Flow-shop Problem

Jerzy Józefczyk and Michał Ćwik

Faculty of Computer Science and Management, Wroclaw University of Technology,

Wybrzeze Wyspianskiego 27, Wroclaw, Poland

Keywords: Decision Making, Optimization, Uncertainty, Interval Data, Minmax Regret, Heuristic Algorithms.

Abstract: A complex population-based solution algorithm for an uncertain decision making problem is presented. The

uncertain version of a permutation flow-shop problem with interval execution times is considered. The

worst-case regret based on the makespan is used for the evaluation of permutations of tasks. The resulting

complex minmax combinatorial optimization problem is solved. The heuristic algorithm is proposed which

is based on the decomposition of the problem into three sequential sub-problems and employs a paradigm of

evolutionary computing. The proposed algorithm solves the sub-problems sequentially. It is compared with

the fast middle point heuristic algorithm via computer simulation experiments. The results show the

usefulness of this heuristic algorithm for instances up to five machines.

1 INTRODUCTION

Investigation of uncertain versions of decision

making problems has a long history. Such problems

being closer to real-world applications are more

complex and difficult to solve. Three issues (Is) are

crucial when considering uncertain problems: the

representation of uncertainty (I1), the evaluation of

arisen uncertain decision making problems (I2), and

the determination of corresponding solution

algorithms (I3). A variety of approaches are

presented and discussed in the literature on all

mentioned issues. Their particular combinations lead

to plenty specific and mostly difficult complex

decision making problems. It is worth noting when

referring to issue I1 that representations of

uncertainty in the form of probability distributions

and the ones based on fuzzy sets and logic seem to

be predominant (Dutt and Kurian, 2013; Aayyub and

Klir, 2006; Klir, 2006). It is assumed that a

probability distribution exists over the space of all

values of corresponding random variables. This

representation is treated as the objective one as the

probability distribution and derivative descriptions

can be empirically verified. Substantial difficulties

with the determination and (or) the estimation of

mentioned probabilistic descriptions, which require

considerable and credible empirical data, is the main

disadvantage of this popular representation. Other

important drawbacks are discussed in (Kouvelis and

Yu, 1997). For the fuzzy approach, the availability

of experimental data on the uncertainty can be

replaced by experts’ opinions in the form of

corresponding membership functions. In a

consequence, the quality of this subjective

representation as well as of following activities

based on it strongly depends on an expert’s quality.

Mentioned shortcomings of both popular

representations have motivated many researchers to

develop other approaches which cope with the

uncertainty more adequately. One of such an

approach is used in the paper. The main idea of the

evaluation of uncertainty as the second mentioned

issue (I2) consists first of all in the substantiation

(determinization) of the uncertainty. The main idea

of such a substantiation consists in transformation of

an uncertain problem into its deterministic

counterpart. Taking the expected value for the

probabilistic representation, when selected

probabilistic distribution reflects the uncertainty, is a

good example of the substantiation. Other frequently

used operators of the substantiation can be found

e.g. in (Yager, 1988). Having the deterministic

problem as the result the issue I3 arises. In fact, all

feasible solution methods and resultant solution

algorithms can be considered, i.e. exact,

approximate as well as heuristic ones.

It is obvious that the choice and justification of

the formalization approach and the solution

algorithm for an unceratin decision making problem

Józefczyk, J. and

Cwik, M.

Heuristic Algorithm for Uncertain Permutation Flow-shop Problem.

In Proceedings of the 1st International Conference on Complex Information Systems (COMPLEXIS 2016), pages 119-127

ISBN: 978-989-758-181-6

119

depend strongly on the problem per se as well as its

complexity and computational difficulty.

Considerations in the paper are limited to the area of

uncertain combinatorial optimization problems with

the parametric uncertainty. It means that for a

combinatorial optimization problem not all

parameters are known, precise, evident or given. The

permutation flow-shop with unlimited buffers to

minimize the makespan is considered, e.g. (Pinedo,

2008). It is one of the most important task

scheduling problems with many applications mainly

in manufacturing, production, logistic and service

systems, but also in information and computing

systems. Generally speaking, the problem deals in

the execution of a finite number of complex tasks by

a finite number of executors (machines, processors).

Each task requires the carrying-out of the same

number of operations, being parts of tasks, which

equals the number of executors. The exact mapping

of executors to operations within tasks is given. A

permutation of tasks is sought minimizing defined

criterion for a given execution times of operations

by corresponding executors. The completion time of

the last task last operation in the permutation plays

often a role of such criterion. An assembly process

of a product performed along a production line is a

good example of the investigated flow-shop

problem. Then, the order of carried-out products

should be determined to minimize the total

production time. The non-deterministic versions of

flow-shop without full information on execution

times are also a subject of many research works, e.g.

(Pinedo and Schage, 1982; Kouvelis et al, 2000;

Averbakh, 2006; Kasperski and Zielinski, 2008).

A specific junction of the mentioned three issues

I1, I2 and I3 is proposed in the paper to solve the

uncertain optimal decision making problem

(optimization problem). Namely, it is assumed that

execution times of tasks by machines are uncertain

(not fully known). However, the information on their

ranges in the form of intervals is only given. The

uncertainty in execution times cause

straightforwardly the uncertainty of the criterion

being the deterministic evaluation of the flow-shop

problem considered. The regret based approach is

proposed to make possible the evaluation of

resultant optimization problem. The notion ‘regret’

assesses the difference between the value of criterion

for fixed realization (scenario) of uncertain

parameters and the optimal value of criterion – for a

given decision (optimization variable). The

application of the regret based approach is

recommended for the interval uncertainty (Kouvelis

and Yu, 1997; Aissi et al, 2009), however, resulting

deterministic combinatorial optimization problem is

extremely complex and difficult. As it has been

pointed out, the regret requires substantiation of the

criterion evaluating a decision with respect to all

feasible scenarios of uncertainty to have the

deterministic evaluation of a decision. In the paper,

the substantiation via maximization is proposed

which expresses the utmost pessimism od a decision

maker (in fact, a decision algorithm) with respect to

scenarios of uncertainty (execution times for the

considered flow-shop) which can occur but are not

known while making a decision. It leads to worst-

case i.e. robust decisions on the one hand but safe

decisions on the other hand. The solution algorithm

determined on such a basis will perform well

irrespective of the actual scenario of uncertainty.

However, it can work fairly when medium scenarios

of uncertainty will take place. The substantiation via

averaging seems to be more adequate for such cases

which, however, can give poor results for extreme

scenarios of uncertainty. The substantiation via

maximization is used hereinafter.

In the paper, a bespoke hybrid heuristic solution

algorithm is proposed to solve the uncertain problem

(issue I3). As it is presented in following sections,

the consequent unceratin flow-shop is extremelly

difficult combinatorial optimization problem, at least

NP-hard one. The rechearches have been focused on

developing of time-effective solution algorithms

appropriated for real-world applications. A hybrid

heuristic algorithm is the result of presented

investigations. The evolutionary computing as an

important paradigm of the computational

intelligence has been employed as the basis for the

developed algorithm.

The uncertain flow-shop problem has been firstly

stated in (Kouvelis and Yu, 1997). Then it has been

investigated in some works. Its NP-hardness was

proved in (Kouvelis et al, 2000) where a branch-

and-bound algorithm and a heuristic procedure

based on a local improvement were also developed.

Particular attention has been paid in this paper to the

elaboration of approximate and heuristic time-

efficient solution algorithms. Some computational

complexity properties were also investigated in

(Kasperski et al, 2012) for the case with discrete

bounded and unbounded scenario sets. The case of

the problem with only two tasks and m machines

was presented in (Averbakh, 2006) where a linear-

time algorithm is given. The evolutionary heuristic

algorithm for the case of three machines was

considered in (Ćwik and Józefczyk, 2015).

The main contribution of the paper deals with

proposing and experimentally evaluating of a time

COMPLEXIS 2016 - 1st International Conference on Complex Information Systems

120

effective hybrid heuristic solution algorithm for

more than three machines. Former works on minmax

regret problems, in general, and on minmax regret

flow-shop problems, in particular, considered mainly

theoretical issues for special cases, e.g. (Averbakh,

2000; Lebedev and Averbakh, 2006; Conde, 2010;

Volgenant and Duin, 2010; Lu et al, 2012). The

results, which can be found there, do not allow us to

have constructive tools for solving real-world

problems, in general, and permutation flow-shop

problems with interval execution times to minimize

the makespan, in particular. The algorithm presented

in the paper fills this gap for the uncertain flow-shop

and refers to analogous works where time-effective

algorithms for other minmax regret problems with

interval uncertainty are presented, e.g. (Józefczyk,

2008; Józefczyk and Siepak, 2013a,b; Siepak and

Józefczyk, 2014; Averbakh and Pereira,. 2011;

Pereira, 2016)

To sum up, the decision making under

uncertainty is a subject of the paper. In particular,

the complex combinatorial NP-hard optimization

problem is solved via hybrid heuristic population-

based solution algorithm. This algorithm can be

treated as a complex tool of computational

intelligence.

The reminder of the paper is organized as

follows. Deterministic and uncertain versions of the

flow-shop investigated are stated in Section 2 as the

combinatorial optimization problems. The idea for

the solution algorithm together with the detailed

presentation of its three component sub-algorithms

are given in Section 3. The next section presents

results of computer simulation experiments

evaluating the algorithm. Section 5 with conclusions

completes the paper.

2 PROBLEM FORMULATION

The flow-shop task scheduling problem is

investigated with a set

}...,,,,,{=

21 mi

MMMM …M

of m machines which are assigned for performing n

tasks constituting a set of tasks

}...,,,,,{=

21 nj

JJJJ …J

. Each task from the set J

needs to be sequentially carried out by all executors

from the set M. For the permutation version of flow-

shop, which is only investigated in the paper, each

machine executes tasks in the same order. Moreover,

the version with unlimited buffers is considered

which means that the task after being completed by

the current machine can leave it and wait if

necessary for the service by the next machine at the

buffer located there. Let us denote by

execution

times of task j by executor i for

mi ...,,2,1=

and

nj ..,,2,1=

which form the matrix

miij

,...,2,1

][

A part of the task performed by a single machine is

called often an operation. Then the problem consists

in the determination of a permutation of tasks

)...,,...,,,(

21 nj

to minimize the

makespan, i.e. the completion time of the last task

from permutation

by the last machine. The

current element

is the number of task

executed as the jth in turn, and

Π∈

where

the set of all

feasible permutations, i.e.

}},...,,2,1{,,:{ kjnkj

≠∈≠=

. The

makespan

),(

max

as the function of p and

can be written in different ways (e.g. Pinedo, 2008).

However, the following recursive form seems to be

the most popular. Let

),(

be the time

moment when ith machine finishes the execution of

task

],),(,),(max[=),(

,,1,,

ππππ

pCpCppC

jjjj

iiii

−

mi ...,,3,2=

nj ...,,3,2=

(1)

Moreover,

,...,,2,1,=),(

mippC

∑

ππ

(2)

and

....,,2,1,=),(

,1,1

njppC

∑

ππ

(3)

In a consequence,

),(),(

,max

pCpC

. The

deterministic flow-shop task scheduling problem

deals with the determination of such

Π∈

minimize

),(

max

. The optimal permutation

′

as well as the optimal makespan

),()(

maxmax

′

pCpC

is obtained. This problem is

NP-hard for

2>m

. The polynomial optimal

Johnson’s algorithm exists for

2=m

(Garey et al,

1976).

Let assume now that crisp values of execution

times

are not known and given. Instead,

intervals for

with possible values of

corresponding execution times are available as only

information of these times. Namely,

ppppp ≤∈ ],,[

. Now, matrix p called a

Heuristic Algorithm for Uncertain Permutation Flow-shop Problem

121

scenario is an element of the Cartesian product of all

scenarios

],[...],[

pppp ××=P

, i.e.

P∈p

The vagueness of p makes it impossible to directly

use the makespan

),(

max

as the evaluation

function of

. The approach based on regret is

applied. The regret

),(min),(

)(),(

maxmax

σπ

pCpC

Π∈

−=

′

−

(4)

is the difference between the value of makespan and

the optimal value of the makespan for fixed p. It is

calculated for current p and

. The regret requires

substantiation according to scenarios. The worst-

case with respect to p has been chosen in the form of

the maximization which gives the criterion

)],(min),([max)(

maxmax

pCpCz

Π∈

∈

−=

(5)

Consequently, the uncertain version of the flow-shop

task scheduling consists in finding such optimal

permutation

∗

that

⎟

⎠

⎞

⎜

⎝

⎛

−=

∈

)],(min),([maxmin)(

maxmax

σππ

σπ

pCpCz

ΠΠ

(6)

for given M, J,

,..,,2,1,,

mipp

nj ..,,2,1=

3 SOLUTION ALGORITHM

The problem (6) called hereinafter P is NP-hard

even for

2=m

(Lebedev and Averbakh, 2006). It is

easy to see that (6) is composed of three nested

optimization sub-problems:

– SP1: inner minimization of

),(

max

with

respect to permutation

, being the deterministic

flow-shop,

– SP2: maximization with respect to feasible

scenarios p, i.e. searching for so called worst-case

scenario, and at the same time calculating the value

of criterion for SP3,

– SP3: outer minimization with respect to

permutation

The majority of results presented in the

corresponding literature refer to cases where SP1 is

easy, i.e. solvable by polynomial algorithms as well

as worst-case scenarios as the solution of SP2 can be

determined either in an intuitional way or by the

reduction to easy optimization problems.

Nevertheless, it is not a case of this paper’s problem.

Now, the problem (6) is a composition of

individually difficult optimization sub-problems

making P extremely difficult issue. In a

consequence, it is easy to justify the lack of any

approximate algorithm for P (Ćwik and Józefczyk,

2015). So, heuristic algorithms are only possible to

solve problem P in a reasonable time. Such an

heuristic algorithm called ALG is proposed in the

paper. To develop it, some simplifications have been

assumed while solving SP1-SP3. First of all, SP1-

SP3 are solved independently and successively by

corresponding individual sub-algorithms. As the

result, we have admittedly a heuristic algorithm but

working in a reasonable and acceptable time. The

main idea for the algorithm ALG solving P is based

on the assumption that SP3 is solved by known

metaheuristics with the dedicated way of calculating

the value of

)(

being the criterion for the

metaheuristics. Let us present the sub-algorithms

successively.

3.1 Sub-algorithm for SP1

The solution of SP1 is replaced by its lower bound

in the form

)minmin(max

)(

LBmax,

∑∑∑

≠

−

++=

ppp

(7)

where first and third elements of the maximum

reflect respectively the following properties:

– at least one task needs to be processed on all

machines indexed from 1 to k – 1 before the kth

machine starts processing operation, unless

1k =

– after ith machine completes the processing, there is

at least one task that needs to be processed on

machines indexed from k + 1 to m, unless

km=

We can take the highest value with respect to all

machines as the fact that (7) holds for all machines.

The details on this lower evaluation of

)(

max

′

can

be found in (Ćwik and Józefczyk, 2015). Such a

form of the lower bound is used by the procedure

calculating approximate value of (5) in SP2.

3.2 Sub-algorithm for SP2

The purpose for SP2 is to calculate the value of

function z for the worst-case scenario. For many

minmax regret problems, it is enough only to

consider extreme point scenarios while searching the

worst-case scenarios, i.e.:

}),{(,

pji ∈∀

(8)

COMPLEXIS 2016 - 1st International Conference on Complex Information Systems

122

Calculation of

max

(,)Cp

can be easily

substituted with the determination of length of the

longest path between

and

in a directed

graph G (V,A) defined by the solution π where V

and A are set of vertexes and arcs, respectively. Each

vertex

corresponds exactly to an individual

execution time of task

executed by machine i.

Therefore, its weight is equal to

. Set A consists

of arcs connecting vertices corresponding to

subsequent operations of the same task

),(

,1,

ππ

, for 0 < i< m) and of operations

performed subsequently on the same machine

),(

+jj

ππ

, for 1< j< n). Assuming that processing

times of operations are unknown, all possible paths

between

and

represent all possible ways of

calculating

max

(,)Cp

. Therefore, we can associate

each of those paths with a separate scenario. Let us

denote path r as a sequence of vertices that belong to

it. Then, we can construct scenario

as follows:

}.for,

for,:{

rvpp

rvpppp

ijijijij

∉=∧

∈==

(9)

As it has been shown in (Ćwik and Józefczyk,

2015), this way we can limit the search from

checking all

extreme points scenarios to

()

()()

1! 1!

⎛⎞

+−

⎜⎟

−−

⎝⎠

paths that exist in G. Then, a simple

enumeration has been applied for such decreased

number of paths, which consists in checking all

feasible paths. This approach turned out effective

only for three machines (

3=m

). Therefore, the

heuristic time-effective procedure is now proposed

to enable its applicability for more than three

machines. The idea is as follows. For each vertex

in graph G, we construct a path from vertex

Obviously, there is only one possible path for i = 1 or

j = 1. Therefore, the construction is trivial. For i > 1

and j > 1, we can see that the path to

needs to

contain one of the two vertices:

−

Assuming that the path to each of them has already

been constructed, we add the last vertex

and

choose the path which yields a greater value of z as a

constructed path for vertex

. After processing all

the vertexes in such a manner, the procedure returns

constructed path for

together with the

corresponding value of z denoted as

. In a

consequence, sub-algorithm for SP2 comprising this

procedure is summarized by the following pseudo-

code.

Input: Graph G(V,A) generated by

feasible solution

Output: Heuristic value of

)(

denoted

)(

1: Set empty matrix cp

2: for

≤

3: for

mj ≤

4: if i==1 or j==1

5: cp[i][j]:= p_r(v[1][1],v[i][j])

6: else:

7: prev1 := cp[i-1][j]

8: prev2 := cp[i][j-1]

9: c1 := prev1 + v[i][j]

10: c2 := prev2 + v[i][j]

11: if z(c1)>z(c2):

12: cp[i][j]:=c1

13: else:

14: cp[i][j]:=c2

15: endif

16 endif

17: endfor

18: endfor

19: return

])][[(

mncpz

3.3 Sub-algorithm for SP3

A standard simple evolutionary algorithm has been

applied for solving SP3 as it has been presented in

(Ćwik and Józefczyk, 2015).

The values of permutation

play directly a role

of a chromosome. Due to the complexity of

()z

the outcome of the sub-algorithm for SP2 in the

form of

)(

is used as the fitness function.

The initial population is constructed via the

random generation of N permutations

(chromosomes). The solution of the middle interval

metaheuristics (MIH) is the basis for generation of a

half of the population. Namely, a MIH-based

permutation undergoes (N/2) – 1 independent

random mutations to have in total (N/2)–element

part of the initial population. The remainder of the

initial population is filled with random permutations

generated uniformly from the search space. MIH

consists in the brute conversion of the uncertain

interval problem into its deterministic counterpart by

assuming the middles of intervals as deterministic

processing times, i.e.

2/)(

MIH

ppp +=

The order crossover operator (Goldberg, 1989)

has been employed to avoid the generation of

Heuristic Algorithm for Uncertain Permutation Flow-shop Problem

123

unfeasible permutations, so no repair algorithm is

required. This operator is characterized by parameter

cross

being the probability of crossing over two

selected chromosomes. Accordingly, 1 –

cross

is the

probability of passing the two chromosomes to the

next generation without any changes. The value of

parameter requires tuning to ensure the best

algorithm performance.

A simple mutation operator has been also used.

Firstly, it is randomly determined if the chromosome

undergoes the mutation. The probability of the

mutation

mut

is considered another parameter of the

algorithm which is tuned. If the chromosome is

decided to undergo the mutation, two random genes

are swapped.

All chromosomes from the population are

selected to generate a new population. The

population is sorted according to decreasing values

of the fitness function. First two chromosomes are

removed from the list and the result of their

crossover is added to the new population. This

process is repeated until the list is empty.

The algorithm terminates when after 5

subsequent iterations no correction is being observed

between the best chromosomes from each

population. The best chromosome of the last

generation is returned as the solution

. The sub-

algorithm for SP3 is presented in the form of

corresponding pseudo-code.

Input: Matrices

and

of size

containing respectively lower and upper

bounds of processing times.

Output: Permutation

and value of

criterion

)

(

1: pop = Initialpopulation()

2: SortByFitFunc(pop)

3: repeat

4: nextgen = list of chromosomes

5: add(nextgen,pop[0])

6: add(nextgen,pop[1])

7: remove(pop,pop[0])

8: remove(pop,pop[1])

9: repeat:

10: ind=RemvoveFromPop(pop[0],pop[1])

11: Sibs = Crossover(ind[1],ind[2])

12: Mutate(Sibs[1])

13: Mutate(Sibs[2])

14: add(nextgen,Sibs[1])

15: add(nextgen,Sibs[2])

16: until(pop is empty)

17: pop = nextgen

18: SortByFitFunc(pop)

19: until (Stop condition is fulfilled)

20: return(pop[0],

([])



zpop0

)

The algorithm for P referred to as ALG obtained

after merging the described sub-algorithms solving

SP1-SP3 is in fact hybrid heuristic population-based

one.

4 NUMERICAL EVALUATION

OF THE ALGORITHM

The algorithm proposed is evaluated via computer

simulation experiments performed using a PC with

Intel Core i5 CPU processor of 2,53 GHz with 4GB

of RAM. According to our best knowledge, there are

no benchmarks in the literature for this problem

which could be used for the comparison. Therefore,

the own instances have been generated in the

following way. For each of mn uncertain parameters

(execution times), the lower bound

,ij

is randomly

generated from the finite interval [0,K] with the

uniform distribution. The upper bound

,ij

is then

generated from the finite interval

ij ij

⎡⎤

⎣⎦

also

with the uniform distribution where K and C are

parameters of the experiments. The default values of

algorithm parameters and problem parameters have

been assumed as:

85.0

cross

15.0

mut

, N = 20,

K = 100, C = 200.

Values of criterion (6) as well as run times of

algorithms are the basis for evaluation. The

algorithm ALG is compared with the heuristic

deterministic algorithm MIH. As it has been

mentioned, MIH generates a solution to any interval

data uncertain problem by solving the deterministic

counterpart which is obtained by substituting all

interval execution times of the problem with its

interval middle points. The arisen deterministic

problem is also NP–hard. Therefore, the NEH

algorithm is applied as the solution tool (Nawaz et

al, 1983). The solution of deterministic problem

MIH

is returned as the solution of the uncertain

problem together with the heuristic value of the

criterion

)(

MIH

calculated by the sub-algorithm

solving SP2. The results are presented in Tables 1–

3. Every numerical result in tables is the mean value

of five independent instances randomly generated

for fixed n and m from given matrices

and

the bounds of interval processing times. Due to

randomness of the evolutionary algorithm each

instance is additionally repeated for this algorithm

five times and the mean result is taken for every of

five instances. In a consequence, every result of

COMPLEXIS 2016 - 1st International Conference on Complex Information Systems

124

Table 1: Computational results for m = 3.

MIH

()z



)

(

MIH

5 447.4 351.36 <0.001 0.309

6 372.2 318.96 0.001 0.402

7 419.8 372.16 0.001 0.600

8 555.0 361.28 0.002 0.785

9 639.0 405.0 0.003 1.093

10 575.6 424.0 0.003 1.361

11 667.6 470.8 0.004 1.294

12 762.2 474.7 0.006 1.724

13 730.8 473.3 0.007 1.892

14 626.6 474.7 0.009 2.085

15 834.8 509.5 0.011 2.459

16 360.4 476.6 0.013 2.913

17 839.4 510.8 0.016 3.643

18 869.2 516.2 0.017 3.812

19 999.6 510.0 0.021 3.876

20 964.2 511.6 0.026 4.089

21 1057.8 522.7 0.029 5.371

22 918.0 538.6 0.032 5.623

23 892.2 548.7 0.037 5.895

24 922.8 522.3 0.040 5.273

25 965.4 537.1 0.044 6.977

26 1207.8 550.9 0.052 6.419

27 886.4 540.6 0.056 8.583

28 1252.8 553.6 0.063 8.548

29 1433.4 559.8 0.070 8.941

30 993.2 564.9 0.079 10.043

Table 2: Computational results for m = 4.

)(

MIH

)

(

MIH

5 488.6 392.4 <0.001 0.594

6 599.6 476.1 <0.001 0.772

7 765.8 643.4 0.002 1.147

8 696.6 605.3 0.002 1.293

9 667.4 548.3 0.003 1.686

10 789.8 650.2 0.004 2.116

11 791.6 621.2 0.005 2.941

12 815.4 702.4 0.007 2.929

13 1076.8 753.5 0.008 3.22

14 906.2 723.2 0.010 3.350

15 903.6 737.0 0.012 4.031

16 1042.6 772.8 0.014 5.109

17 1129.2 808.6 0.018 6.182

18 1173.8 779.4 0.020 6.729

19 1218 812.9 0.023 6.383

20 1242 831.4 0.029 7.312

21 1140.6 820.3 0.033 8.593

22 1336.6 811.2 0.036 10.425

23 1394.8 825.8 0.040 10.433

24 1468.2 847.5 0.045 12.819

25 1252.4 858.4 0.051 12.970

26 1645.8 891.4 0.059 14.852

27 1448.8 881.5 0.064 16.267

28 1255.4 898.4 0.072 18.853

29 1500.8 857.5 0.079 14.276

30 1463.8 856.4 0.087 19.002

Table 3: Computational results for m = 5.

)(

MIH

)

(

MIH

622.4 544.7 <0.001 0.790

742.2 629.8 0.002 0.987

786.6 677.0 0.002 1.619

917.8 790.0 0.003 1.758

848.4 753.2 0.003 2.428

909.4 806.1 0.005 2.607

1032.6 845.7 0.006 3.541

1051.4 879.6 0.007 4.259

1121.8 983.3 0.009 5.238

1073.2 940.6 0.012 5.007

1241.4 1018.6 0.014 5.668

1315.8 1088.4 0.017 7.327

1357.4 1099.3 0.020 9.004

1453.0 1068.6 0.024 10.628

1381.8 1113.3 0.028 12.190

1587.6 1153.0 0.030 12.271

1616.4 1148.0 0.036 16.190

1630.0 1158.2 0.044 18.645

1681.8 1155.0 0.046 18.397

1594.6 1175.2 0.054 17.910

1586.4 1180.0 0.063 20.780

1875.2 1214.9 0.067 23.18

1721.6 1199.9 0.074 24.570

1832.2 1201.7 0.083 26.622

1917.0 1237.2 0.092 32.713

2163.8 1207.3 0.104 40.000

ALG is, in fact, the mean value of 25 independent

random instances. The values of criterion as well as

the run times in seconds for MIH and ALG are

respectively denoted in corresponding columns of

tables as

)(

MIH

and

)

(

as well as

MIH

and T.

The results confirmed the usefulness of ALG in

comparison with MIH from both the criterion

and

the computational time points of view. It is

necessary to point out that the values of criterion (5)

for the resulting permutations

MIH

and

were

calculated by the sub-algorithm SP2, i.e. the values

instead of z are the basis for this comparison. It

turned out that ALG is substantially better than

MIH. Namely, the relative improvement calculated

as the ratio

)

(

)(

MIH

(10)

fluctuates from 1,11 to 1.79 (except the instance for

m = 3 and n = 16 when MIH is better). The mean

improvement for all instances is equal to 1.36. The

computational time of ALG is fully acceptable

however much longer than for MIH.

Heuristic Algorithm for Uncertain Permutation Flow-shop Problem

125

5 CONCLUSIONS

The complex and extremely difficult combinatorial

optimization problem, i.e. the uncertain permutation

flow-shop with the makespan as criterion, has been

investigated. The case of minmax regret with

interval execution times has been considered. The

nested three sub-problems have been solved

independently by heuristic sub-algorithms. The

algorithm of the whole optimization problem as the

consecutive composition of all three sub-algorithms

has been compared with the MIH metaheuristics

which is often applied for minmax regret problems

with interval data. MIH has good properties for

many simple optimization problems. For example, it

is 2-approximate algorithm for the special case of

flow-shop problem considered in the paper with only

two machines (m = 2). Therefore, this metaheuristics

has been chosen as the basis for comparison. It is

worth noting that the calculation of criterion for

SP3, i.e. for the whole problem is also NP-hard. The

comparison uses its approximate value

Unfortunately, the relation between z and

is not

known. This issue requires more profound studies.

The preliminary considerations show the usefulness

as the approximate value of the criterion for

solution

. Namely, four very simple instances

have been considered for m = 3 and 4 as well as

n = 4 and 5. These instances have been solved by

ALG as well as their optimal solutions have been

derived by the brute force algorithm. The relative

differences between both results in the sense of ratio

)(/)

(

ππ

analogous to (10) are presented in

Table 4. The values of criterion

are worse than

no more than 1.30.

The research of this problem is now continuing.

For example, the instances for greater m and n are

solved as well as more effective and faster sub-

algorithm for SP3 is verified.

Table 4: Values of

)(/)

(

ππ

for different m and n.

m\n 4 5

1.21 1.13

1.30 1.14

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