Optimalisasi Scheduling

Abstrak
The prevalent approach to the treatment of processing time uncertainties in production Pendekatan umum untuk pengobatan ketidakpastian waktu proses produksi
scheduling problems is through the use of probabilistic models. penjadwalan masalah adalah melalui penggunaan model probabilistik. Apart from requiring detailed Selain membutuhkan rinci
information about probability distribution functions, this approach also has the drawback that informasi tentang fungsi distribusi probabilitas, pendekatan ini juga memiliki kelemahan yang
the computational expense of solving these models is very high. biaya komputasi memecahkan model-model ini sangat tinggi. In this work, we present a Dalam karya ini, kami menyajikan
non-probabilistic treatment of scheduling optimization under uncertainty, based on concepts non-probabilistik pengobatan optimasi penjadwalan di bawah ketidakpastian, yang berdasarkan pada konsep
from fuzzy set theory and interval arithmetic, to describe the imprecision and uncertainty in dari teori himpunan fuzzy dan aritmatika interval, untuk menggambarkan ketidaktepatan dan ketidakpastian dalam
the task durations. durasi tugas. We first provide a brief review on the fuzzy set approach, comparing it Kami pertama-tama memberikan kajian singkat tentang pendekatan himpunan fuzzy, membandingkannya
with the probabilistic approach. dengan pendekatan probabilistik. We then present MILP models derived from applying this ap- Kami kemudian menyajikan model MILP berasal dari penerapan pendekatan ini-
proach to two different problems - flowshop scheduling and new product development process proach untuk dua masalah yang berbeda - penjadwalan flowshop dan proses pengembangan produk baru
scheduling. penjadwalan. Results indicate that these MILP models are computationally tractable for reason- Hasil penelitian menunjukkan bahwa model ini komputasi MILP penurut karena alasan-
ably sized problems. dengan kemampuan berukuran masalah. We also describe tabu search implementations in order to handle larger Kami juga menjelaskan implementasi pencarian tabu untuk menangani lebih besar
problems. masalah.
1 Introduction 1 Pendahuluan
With increased interest in production scheduling in chemical engineering in recent years, there has Dengan meningkatnya minat dalam penjadwalan produksi di teknik kimia dalam beberapa tahun terakhir, ada
been much work addressing the optimal scheduling of a wide range of sytems - from individual telah banyak pekerjaan yang menangani penjadwalan optimal dari berbagai sistem tersebut - dari individu
production units to geographically distributed multisite supply chains (see Shah, 1998, for review). unit produksi untuk didistribusikan secara geografis rantai pasokan multisite (lihat Shah, 1998, untuk ditinjau ulang).
¢ ¢
jayanth@cmu.edu jayanth@cmu.edu
£ £
grossmann@cmu.edu - Corresponding author grossmann@cmu.edu - Sesuai penulis
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A significant amount of the work in this area has focussed on the development of deterministic Sebuah jumlah yang signifikan dari pekerjaan di daerah ini telah difokuskan pada pengembangan deterministik
models, where the problem data is assumed to be known in advance. model, dimana data masalah diasumsikan diketahui terlebih dahulu. In reality, though, there can Pada kenyataannya, meskipun, tidak
be uncertainty in a number of factors such as processing times and costs. akan ketidakpastian dalam beberapa faktor seperti waktu pemrosesan dan biaya. As a result, a number Sebagai hasil, angka
of papers in the recent years have addressed scheduling in the face of uncertainties in different kertas pada beberapa tahun terakhir telah membahas penjadwalan dalam menghadapi ketidakpastian di berbagai
parameters - for eg, demands (Ierapetritou and Pistikopoulos, 1996; Petkov and Maranas, 1997; parameter - untuk misalnya, tuntutan (Ierapetritou dan Pistikopoulos, 1996; Petkov dan Maranas, 1997;
Sand et al. , 2000) and processing times (Honkomp et al. , 1999; Schmidt and Grossmann, 2000; Pasir et al 1999., 2000) dan pengolahan kali (et Honkomp; al., Schmidt dan Grossmann, 2000;
Balasubramanian and Grossmann, 2001). Balasubramanian dan Grossmann, 2001).
The prevalent approach to the treatment of these uncertainties is through the use of probabilis- Pendekatan umum untuk pengobatan ketidakpastian ini adalah melalui penggunaan probabilis-
tic models that describe the uncertain parameters in terms of probability distributions. tic model yang menggambarkan parameter pasti dalam hal distribusi probabilitas. However, Namun,
the evaluation and optimization of these models is computationally expensive, either because of the evaluasi dan optimalisasi model-model ini komputasi mahal, baik karena
large number of scenarios resulting from a discrete representation of the uncertainty (Wets, 1974), jumlah besar skenario yang dihasilkan dari representasi diskrit dari ketidakpastian (membasahi, 1974),
or the need to use complicated multiple integration techniques when the uncertainty is represented atau kebutuhan untuk menggunakan beberapa teknik integrasi rumit bila ketidakpastian diwakili
by continuous distributions (Schmidt and Grossmann, 2000). oleh distribusi kontinu (Schmidt dan Grossmann, 2000). Furthermore, the use of probabilistic Selain itu, penggunaan probabilistik
models is realistic only when these descriptions of the uncertain parameters is available, say from model yang realistis hanya jika ini deskripsi dari parameter yang tidak pasti tersedia, mengatakan dari
historic data. bersejarah data. When such data is not available (for example in New Product Development, when Bila data tersebut tidak tersedia (misalnya di Pengembangan Produk Baru, ketika
the tasks have never been or only rarely performed before), we do not have enough information for tugas tidak pernah atau hanya jarang dilakukan sebelumnya), kita tidak memiliki informasi yang cukup untuk
inferring or deriving the probabilistic models. menyimpulkan atau berasal model probabilistik. In such situations, we have to resort to an alternative Dalam situasi seperti itu, kita harus resor untuk alternatif
treatment of uncertainty. pengobatan ketidakpastian. For example, the modeler may be able to approximate the duration of the Sebagai contoh, pemodel mungkin dapat perkiraan durasi
tasks and specify the longest and shortest durations or the interval in which the duration belongs at tugas dan menentukan jangka waktu terpendek dan terpanjang atau interval di mana durasi berada di
different levels of confidence. berbagai tingkat kepercayaan.
In this work, we draw upon concepts from fuzzy set theory and interval arithmetic to describe Dalam karya ini, kita memanfaatkan konsep-konsep dari teori himpunan fuzzy dan aritmatika interval untuk menggambarkan
the imprecision and uncertainties in the durations of batch processing tasks. ketidakakuratan dan ketidakpastian dalam jangka waktu tugas batch processing. Indeed, this approach Memang, pendekatan ini
has been receiving increasing attention recently. telah menerima perhatian yang meningkat baru-baru ini. McCahon and Lee (1992) were the first to illus- McCahon dan Lee (1992) adalah yang pertama kali ilus-
trate the application of fuzzy set theory as a means of analyzing performance characteristics for a trate penerapan teori himpunan fuzzy sebagai sarana karakteristik kinerja analisis untuk
flowshop system. flowshop sistem. They modified the Campbell, Dudek and Smith sequencing heuristic (Campbell Mereka memodifikasi Campbell, Dudek dan Smith sequencing heuristik (Campbell
et al. , 1970) for the case of fuzzy processing times. et al)., 1970 untuk kasus kali pengolahan fuzzy. However, their procedure was cumbersome Namun, prosedur mereka praktis
for fairly large problem sizes. untuk masalah ukuran besar cukup. Most of the work in applying fuzzy set theory to scheduling opti- Sebagian besar pekerjaan dalam menerapkan teori himpunan fuzzy untuk penjadwalan Opti-
mization has primarily focussed on using heuristic search techniques such as simulated annealing mization telah terutama difokuskan pada menggunakan teknik pencarian heuristic seperti simulated annealing
and genetic algorithms to obtain near-optimal solutions. dan algoritma genetika untuk mendapatkan solusi optimal dekat. For example, Ishibuchi et al. (1994) ex- Sebagai contoh, Ishibuchi et al ex. (1994)-
amined flowshop scheduling problems with fuzzy due dates, where the membership function of amined masalah penjadwalan flowshop dengan tanggal jatuh tempo fuzzy, dimana fungsi keanggotaan
the fuzzy due date indicates the grade of satisfaction of the decision maker with the completion tanggal jatuh tempo fuzzy menunjukkan tingkat kepuasan dari pembuat keputusan dengan penyelesaian
time of a job. waktu pekerjaan. These authors presented results from applying genetic algorithms to two types of Para penulis ini mempresentasikan hasil dari penerapan algoritma genetika untuk dua jenis
problems - i) maximizing the minimum grade of satisfaction over all jobs, and ii) maximizing the masalah - i) memaksimalkan nilai minimum dari kepuasan atas semua pekerjaan, dan ii) memaksimalkan
total grade of satisfaction. kelas total kepuasan. Fortemps (1997) addressed the problem of minimizing the makespan of Fortemps (1997) membahas masalah meminimalkan makespan dari
jobshops with fuzzy durations using simulated annealing. jobshops dengan jangka waktu fuzzy menggunakan simulated annealing. The author also introduced a new com- Penulis juga memperkenalkan com baru
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parison method for fuzzy numbers. perbandingan metode untuk nomor fuzzy. A recent paper (Ozelkan and Duckstein, 1999) investigates Sebuah paper baru-baru ini (Ozelkan dan Duckstein, 1999) menginvestigasi
the necessary conditions for optimality of fuzzy counterparts of classical (deterministic) optimal yang diperlukan kondisi untuk optimalisasi rekan fuzzy klasik (deterministik) optimal
scheduling rules and emphasizes the importance of using ranking functions that satisfy certain penjadwalan aturan dan menekankan pentingnya menggunakan fungsi yang memenuhi peringkat tertentu
properties. properti. In the area of project scheduling, Chanas and Kamburowski (1981) laid the theoretical Di bidang penjadwalan proyek, Chanas dan Kamburowski (1981) meletakkan teori
foundations for approaching the problem through the fuzzy set framework. fondasi untuk mendekati masalah melalui kerangka himpunan fuzzy. Lootsma (1989) con- Lootsma (1989) con-
trasted the fuzzy set approach with the well-known stochastic PERT (Malcolm et al. , 1959) and trasted pendekatan himpunan fuzzy dengan dikenal dengan baik stokastik PERT (et al Malcolm 1959.,) dan
discussed some of the pitfalls associated with stochastic PERT, including the computational com- dibahas beberapa jebakan yang terkait dengan PERT stokastik, termasuk komputasi com-
plexity of PERT (Hagstrom, 1988). kerumitan dari PERT (Hagstrom, 1988). Hapke and Slowinski (1996) and Wang (1999) addressed the Hapke dan Slowinski (1996) dan Wang (1999) membahas
resource-constrained project scheduling problem. sumber daya terbatas masalah penjadwalan proyek. While the former applied heuristic dispatching Sementara mantan diterapkan heuristik dispatching
rules to generate the schedules, the latter formulated the problem as a fuzzy constraint satisfaction aturan untuk menghasilkan jadwal, yang kedua dirumuskan masalah sebagai kepuasan kendala fuzzy
problem based on possibility theory (Dubois and Prade, 1988) and used a beam search (Ow and masalah berdasarkan teori kemungkinan (Dubois dan Prade, 1988) dan menggunakan pencarian balok (Ow dan
Morton, 1988) to obtain a schedule with the least possibility of being late. Morton, 1988) untuk memperoleh jadwal dengan kemungkinan sedikit terlambat. We refer the reader to Kita mengacu pembaca
Slowinski and Hapke (2000) as an excellent overview of scheduling under fuzziness. Slowinski dan Hapke (2000) sebagai gambaran yang sangat baik dari penjadwalan bawah ketidakjelasan. In the chem- Dalam chem-
ical engineering literature, the theory of fuzzy sets has primarily been applied for process control literatur rekayasa ical, teori fuzzy set terutama yang telah diterapkan untuk pengendalian proses
(see for examples of recent works, Postlethwaithe and Edgar, 2000; Galluzo et al. , 2001) although (Lihat contoh karya terbaru, Postlethwaithe dan Edgar, 2000; dkk Galluzo 2001.,) Walaupun
there have also been applications in the area of design (Kubic and Stein, 1988; Tarifa and Chiotti, ada juga aplikasi dalam bidang desain (Kubic dan Stein, 1988; Tarifa dan Chiotti,
1995). 1995). Majozi and Zhu (2001) recently presented an application of fuzzy set theory to the optimal Majozi dan Zhu (2001) baru-baru ini disajikan sebuah aplikasi dari teori himpunan fuzzy untuk yang optimal
allocation of operators to batch plants. alokasi operator untuk tanaman batch.
Unlike the previously mentioned papers, which have relied on heuristics to obtain near- Berbeda dengan yang disebutkan kertas sebelumnya, yang mengandalkan heuristik untuk mendapatkan dekat-
optimal solutions, we show how it is possible to develop Mixed Integer Linear Programming solusi optimal, kami tunjukkan bagaimana mungkin untuk mengembangkan Mixed Integer Linear Programming
(MILP) models for different scheduling problems that use a fuzzy representation of uncertainty. (MILP) model untuk masalah penjadwalan yang berbeda yang menggunakan representasi fuzzy ketidakpastian.
We also compare the MILP approach with heuristic techniques for flowshop problems. Kami juga membandingkan pendekatan MILP dengan teknik heuristik untuk masalah flowshop. As a brief Sebagai singkat
introduction to the concepts of this non-probabilistic approach, we present results from flowshop pengenalan konsep ini-pendekatan non probabilistik, kami menyajikan hasil dari flowshop
optimization under uncertainty. optimasi di bawah ketidakpastian. The durations of the tasks are described by fuzzy numbers (as Durasi tugas yang digambarkan oleh angka fuzzy (sebagai
opposed to deterministic values or probabilistic descriptions). bertentangan dengan nilai-nilai deterministik atau deskripsi probabilistik). We present MILP models for se- Kami menyajikan model MILP untuk se-
lecting the optimal schedule for two types of flowshop systems - (i) multistage flowshops with a lecting jadwal optimal untuk dua jenis sistem flowshop - (i) multistage flowshops dengan
single processing unit in each stage (Birewar and Grossmann, 1989; Pekny and Miller, 1991), and tunggal pengolahan unit di setiap tahap (Birewar dan Grossmann, 1989; Pekny dan Miller, 1991), dan
(ii) multistage flowshops with multiple parallel units in each stage (Pinto and Grossmann, 1995). (Ii) multistage flowshops dengan unit paralel pada setiap tahap (Pinto dan Grossmann, 1995).
We then address the problem of scheduling the New Product Development (NPD) process for an Kami kemudian mengatasi masalah penjadwalan Pengembangan Produk Baru (NPD) proses untuk
agricultural chemical under uncertainty (Schmidt and Grossmann, 1996). kimia pertanian di bawah ketidakpastian (Schmidt dan Grossmann, 1996). In the NPD process, the Dalam proses NPD, yang
potential product must pass a series of tests that assess its safety, efficacy and environmental im- potensi produk harus melewati serangkaian tes yang menilai keamanan, efikasi dan lingkungan im-
pact. pakta. The costs, durations and probabilities of success of various testing tasks are often not known Biaya, waktu dan probabilitas keberhasilan tugas berbagai pengujian sering tidak diketahui
with certainty when the schedule has to be constructed. dengan pasti ketika jadwal harus dibangun. We consider the problem of optimizing Kami mempertimbangkan masalah optimalisasi
the schedule of testing tasks while taking into account the uncertainties in the durations and suc- jadwal pengujian tugas dengan memperhatikan ketidakpastian dalam jangka waktu dan SUC-
cess of the tasks, a problem which has not been reported before. cess tugas, masalah yang belum pernah dilaporkan sebelumnya. The optimization of the testing Optimasi pengujian
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schedule consists in introducing precedence constraints between the tests, in addition to the given jadwal terdiri dalam memperkenalkan kendala prioritas antara tes, di samping yang diberikan
set of technological precedence constraints. set kendala prioritas teknologi. We present an MILP formulation for the objective of Kami menyajikan formulasi MILP untuk tujuan
maximizing the difference of the expected value of the fuzzy income and the expected cost of a memaksimalkan perbedaan nilai yang diharapkan dari pendapatan fuzzy dan biaya yang diharapkan dari
testing schedule. jadwal pengujian.
The paper is organized as follows. Makalah ini disusun sebagai berikut. Section 2 defines the problems of interest. Bagian 2 mendefinisikan masalah bunga. Section 3 Bagian 3
presents a brief overview of fuzzy set theory and compares mathematical operations in the fuzzy menyajikan gambaran singkat mengenai teori himpunan fuzzy dan membandingkan operasi matematika dalam fuzzy
framework with their probabilistic counterparts. kerangka probabilistik dengan rekan-rekan mereka. After an overview of our approach to the problem Setelah ikhtisar pendekatan kita terhadap masalah
in Section 4, we present results from optimization of flowshops in Sections 5 and 6 and the NPD dalam Bagian 4, kami menyajikan hasil dari optimasi flowshops dalam Bagian 5 dan 6 dan NPD
process in Section 7. proses dalam Bagian 7. Finally, the conclusions on this work are drawn. Akhirnya, kesimpulan pada pekerjaan ini diambil.
2 Problem Statement 2 Pernyataan Masalah
We consider in this paper the following general problem. Kami menganggap dalam makalah ini masalah umum berikut. Given are a set of tasks (orders or tests) Mengingat adalah seperangkat tugas (perintah atau tes)
to be performed using certain resources (processing equipment). untuk dilakukan dengan menggunakan sumber daya tertentu (peralatan pemrosesan). These tasks have their durations Tugas ini memiliki jangka waktu yang mereka
specified as fuzzy numbers. ditetapkan sebagai bilangan fuzzy. The objective is to determine an allocation of the resources to the Tujuannya adalah untuk menentukan alokasi sumber daya ke
tasks that is optimal with respect to a certain measure such as the makespan or the profit. tugas yang optimal sehubungan dengan ukuran tertentu seperti makespan atau keuntungan. Our Kami
principal objective is to demonstrate the application of concepts from fuzzy set theory to scheduling Tujuan utama adalah untuk menunjukkan penerapan konsep-konsep dari teori himpunan fuzzy untuk penjadwalan
optimization when the processing times are uncertain. optimasi ketika waktu pengolahan tidak pasti. We present applications from flowshop and Kami menyajikan aplikasi dari flowshop dan
NPD testing optimization. NPD pengujian optimasi.
3 Comparison of Fuzzy Sets and Numbers with Probabilistic 3 Perbandingan Fuzzy Set dan Nomor dengan Probabilistik
Approach Pendekatan
In this section, we review key concepts from the theory of fuzzy sets that will be used for the Pada bagian ini, kami meninjau konsep-konsep kunci dari teori fuzzy set yang akan digunakan untuk
scheduling models. penjadwalan model. We also compare them with their probabilistic counterparts. Kami juga membandingkannya dengan rekan probabilistik mereka.
3.1 Definitions 3.1 Definisi
A classical set Satu set klasik
¤ ¤
of a universe sebuah alam semesta
¥ ¥
is a collection of elements or objects that are well defined and adalah kumpulan elemen atau objek yang didefinisikan dengan baik dan
possess some common properties. memiliki beberapa sifat umum. An element Sebuah elemen
¦ |
of the universe alam semesta
¥ ¥
may either belong to ( true or 1) mungkin baik milik (true atau 1)
¤ ¤
or not ( false or 0). atau tidak (palsu atau 0). Zadeh (1965) introduced the concept of a fuzzy set in which the membership Zadeh (1965) memperkenalkan konsep tentang himpunan fuzzy di mana keanggotaan
of an element to a set need not be just binary-valued (ie, 0-1), but could be any value over the suatu unsur untuk menetapkan tidak perlu hanya bernilai biner (yaitu, 0-1), tetapi bisa nilai apapun atas
interval [0,1] depending on the degree to which the element belongs to the set. interval [0,1] tergantung pada sejauh mana elemen milik mengatur. The higher the Semakin tinggi
degree of belonging, higher the membership value. derajat milik, semakin tinggi nilai keanggotaan.
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A fuzzy set Sebuah himpunan fuzzy
§ §
ö ö
the universe alam semesta
¥ ¥
is specified by a membership function ditentukan oleh suatu fungsi keanggotaan
© ¦ © |
which takes yang mengambil
its value in the interval [0,1]. nya nilai dalam] interval [0,1. For each element Untuk setiap elemen
¦ |
of dari
¥ ¥
, the quantity , Kuantitas
© ¦ © |
specifies the degree to menentukan derajat
which yang
¦ |
belongs in . termasuk dalam. Thus, Dengan demikian,
§ §
ï ï
completely characterized by the set of ordered pairs: sepenuhnya ditandai oleh serangkaian pasangan memerintahkan:
§ §
¦$#%©'& |$#%©'&
¦ ( 0)1¦324¥65 | (0) 1 | 324 ¥ 65
(1) (1)
The support of a fuzzy set Dukungan dari himpunan fuzzy
§ §
ï ï
a (crisp/classical) subset of X given by a (garing / klasik) subset dari X diberikan oleh
798A@B@ 798A @ B @
C§ C §
"! "!
¦D24¥ | ¥ D24
)E©'& ) E © '&
¦ GFIHP5 | GFIHP5
(2) (2)
The The
Q Q
-level set ( -Level set (
Q Q
-cut) of a fuzzy set -Cut) dari sebuah himpunan fuzzy
§ §
ï ï
a crisp subset of X given by subset garing X diberikan oleh
§ §
0 0
¦D2T¥ | D2T ¥
)U©'& ) U © '&
¦ GVWQX5 | GVWQX5
Y`Qa2b cHd#9egf Y `Qa2b CHD # 9egf
(3) (3)
A fuzzy number is defined as a bounded support fuzzy set of the real line Sebuah bilangan fuzzy didefinisikan sebagai fuzzy set dukungan terbatas dari garis riil
h h
whose yang
Q Q
-cuts are -Pemotongan yang
closed intervals. interval tertutup.
3.2 Arithmetic Operations - a Comparison with Probabilistic Computations 3.2 Operasi Aritmatika - sebuah Perbandingan dengan Komputasi Probabilistik
Arithmetic operations on fuzzy numbers that will be used for the objective function and the con- Operasi aritmatika pada bilangan fuzzy yang akan digunakan untuk fungsi objektif dan con-
straints, such as addition, subtraction, taking the maximum of two fuzzy numbers are defined kendala, seperti penambahan, pengurangan, mengambil maksimal dua nomor fuzzy didefinisikan
through the extension principle (Zadeh, 1965), which allows the extension of operations on real melalui prinsip ekstensi (Zadeh, 1965), yang memungkinkan perpanjangan operasi pada nyata
numbers to fuzzy numbers. nomor ke nomor fuzzy. In the scheduling problems of interest to us, the principal arithmetic Dalam masalah penjadwalan yang menarik bagi kita, aritmatika pokok
operations that are involved are addition (computing the fuzzy end-time of a task given the fuzzy operasi yang terlibat adalah tambahan (komputasi akhir zaman fuzzy tugas diberi fuzzy
start time and duration) and maximization (computing the start time of a task as the maximum of waktu mulai dan durasi) dan maksimalisasi (menghitung waktu mulai tugas sebagai maksimum
the end times of preceding tasks). akhir zaman tugas sebelumnya). Hence, we restrict our description of arithmetic operations on Oleh karena itu, kami membatasi deskripsi kita operasi aritmatika pada
fuzzy numbers to these two operations. fuzzy angka untuk kedua operasi.
3.2.1 Addition 3.2.1 Penambahan
1. 1. Fuzzy Numbers Bilangan Fuzzy
If Jika
§ §
¥ ¥
and dan
§ §
i i
are two fuzzy numbers, their addition can be accomplished by using adalah dua bilangan fuzzy, selain mereka dapat dicapai dengan menggunakan
Q Q
-level cuts. Tingkat pemotongan.
If Jika
§ §
¥ ¥
R' qp R 'QP
¦sr | Sr
R R
#(¦ut # (| Ut
R R
f f
and dan
§ §
i i
R' vpxw R 'vpxw
r r
R R
# #
w w
t t
R R
f f
, then the result , Maka hasilnya
§ §
y y
which is the addition of yang merupakan penambahan
§ §
¥ ¥
and dan
§ §
i i
can bisa
be defined by the didefinisikan oleh
Q Q
-level sets as below: Tingkat set sebagai berikut:
§ §
y y
R€ R €
§ §
¥ ¥
R R
c ‚ u§ c, u §
i i
RS vp RS vp
¦ |
r r
R R
w w
r r
R R
#(¦ # (|
t t
R R
w w
t t
R R
f f
Y`Qb2a cHƒ#9e„f Y `Qb2a cHƒ # 9e" f
(4) (4)
This clearly shows that the lower bound of Hal ini jelas menunjukkan bahwa batas bawah
§ §
y y
is the sum of the lower bounds of adalah jumlah batas bawah
§ §
¥ ¥
and dan
§ §
i i
and similarly, the upper bound of dan juga, batas atas
§ §
y y
is the sum of the upper bounds of adalah jumlah batas atas
§ §
¥ ¥
and dan
§ §
i i
. . Figure 1 Gambar 1
provides an example. memberikan contoh.
5 5
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10 10
35 35
t t
m m
X X
(t) (T)
20 20
1.0 1.0
0 0
5 5
25 25
t t
30 30
1.0 1.0
0 0
1.0 1.0
15 15
65 65
t t
45 45
0 0
Y Y
X X
Z Z
Figure 1: Gambar 1:
Addition of TFNs Penambahan TFNs
2. 2. Random Variables Random Variabel
If Jika
¥ ¥
and dan
i i
are two independent random variables with probability density functions ( dua variabel acak independen dengan fungsi densitas probabilitas (
…‡†‰ˆ ... ‡ † ‰
s) s)
given by diberikan oleh
U'' “” U''""
and dan
U•– “” U • - ""
, and , Dan
y y
is a random variable that is the sum of adalah variabel acak yang adalah jumlah dari
¥ ¥
and dan
i i
, the , Yang
…‡†—ˆ ... ‡ † -
of dari
y y
is obtained by convolving the distribution of diperoleh dengan convolving distribusi
¥ ¥
with that of dengan yang
i i
. . As is well known, the Sebagaimana telah diketahui,
expected value of nilai harapan
y y
is given by the addition of the expected values of diberikan dengan penambahan nilai-nilai yang diharapkan
¥ ¥
and dan
i i
. .
y y
¥˜ ¥ ~
i i
d™ “” d ™ ""
1'e “” fg U•h “” 1'e "" fg • h U ""
jilknm jilknm
o o
m m
1'e 1'e
8 8
pE U•h “ q pE U • h "q
8 8
pUr pur
8 8
(5) (5)
s s
py py
f f
s s
p p
¥4fP ¥ 4fP
s s
pi pi
f f
(6) (6)
3.2.2 Maximum Operator 3.2.2 Operator Maksimum
1. 1. Fuzzy Numbers Bilangan Fuzzy
If Jika
§ §
¥ ¥
and dan
§ §
i i
are two fuzzy numbers, their maximum can also be obtained by using adalah dua bilangan fuzzy, maksimum mereka juga dapat diperoleh dengan menggunakan
Q Q
-level Tingkat
cuts. pemotongan. Thus, Dengan demikian,
§ §
y y
R€ tluEv R € tluEv
w§ w §
¥ ¥
R R
# § # §
i i
R R
xpxtluEv xpxtluEv
¦ |
r r
R R
# #
w w
r r
R R
y# y #
tluEv tluEv
¦ |
t t
R R
# #
w w
t t
R R
zf zf
Y`Q{2b |Hƒ#9egf Y `Q {2b | Hƒ # 9egf
(7) (7)
In general, this operation requires infinite computations, ie, evaluating maxima for every Secara umum, operasi ini memerlukan perhitungan yang tak terbatas, yaitu, evaluasi maxima untuk setiap
Q}2 |Hƒ#9egf Q} 2 | Hƒ # 9egf
. . However, good approximations can be obtained by performing these compu- Namun, pendekatan yang baik dapat diperoleh dengan melakukan ini compu-
tations at specific values of Kumai sebesar nilai tertentu
Q Q
, rather than at all values. , Bukan di semua nilai. As can be seen in Figure 2, the Seperti dapat dilihat pada Gambar 2,
maximum of two TFNs need not be another TFN. maksimal dua TFNs tidak perlu TFN lain.
2. 2. Random Variables Random Variabel
If Jika
¥ ¥
and dan
i i
are two independent random variables with dua variabel acak independen dengan
…‡†‰ˆ ... ‡ † ‰
s given by s yang diberikan oleh
1'‚ “” 1 ', ""
and dan
U• “” U • ""
, ,
and dan
y y
is a random variable that is the maximum of adalah variabel acak yang maksimum
¥ ¥
and dan
i i
, the cumulative distribution , Distribusi kumulatif
function ( fungsi (
€ †‰ˆ € † ‰
) of ) Dari
y y
is computed by multiplying the dihitung dengan mengalikan
€‡†—ˆ € ‡ † -
of dari
¥ ¥
with the dengan
€‡†—ˆ € ‡ † -
of dari
i i
. .
Unlike the case of the addition operation, the expected value of the maximum of two random Tidak seperti kasus operasi penjumlahan, nilai yang diharapkan dari maksimal dua acak
6 6
Page 7 Page 7
Figure 2: Gambar 2:
Maximum of TFNs Maksimum TFNs
variables cannot be obtained by taking the maximum of the expected values. variabel tidak dapat diperoleh dengan mengambil nilai maksimum dari nilai-nilai yang diharapkan. In fact, taking Bahkan, mengambil
the maximum of the expected values results in a lower bound to the actual value. maksimum hasil nilai-nilai yang diharapkan dalam batas bawah dengan nilai yang sebenarnya.
y y
tluEv tluEv
¥4# 4 ¥ #
i i
ˆ ™C “” ™ C ""
ˆ$'‚ “” pUˆC• “” $ ', "" PUC • ""
(8) (8)
s s
pƒ‚…„ pƒ, ... "
¦ ¥4# | ¥ 4 #
i i
†f‡V F † ‡ V
‚…„ , ... "
¦ |
ps id
¥ˆ ‰# ¥ ‰ #
s s
i i
zf zf
(9) (9)
3.3 Objective Function Comparisons 3.3 Perbandingan Fungsi Tujuan
3.3.1 Fuzzy Numbers 3.3.1 Bilangan Fuzzy
Since we are interested in selecting a schedule with the optimum (minimum) makespan, we have Karena kita tertarik dalam memilih jadwal dengan makespan (minimum) yang optimal, kami telah
to compare the fuzzy makespans of potential schedules. untuk membandingkan makespans fuzzy jadwal potensial. In a probabilistic treatment, we may be Dalam pengobatan probabilistik, kita mungkin
interested in minimizing the expected value of the makespan. tertarik dalam mengurangi nilai yang diharapkan dari makespan. However, with our approach, matters Namun, dengan pendekatan kami, hal-hal
are not so straightforward. tidak begitu jelas.
There exists a large body of literature that deals with the comparison of fuzzy numbers (see Terdapat tubuh besar literatur yang berhubungan dengan perbandingan bilangan fuzzy (lihat
Chen and Hwang, 1992, for an overview) and a number of indices have been proposed. Chen dan Hwang, 1992, untuk ikhtisar) dan sejumlah indeks telah diusulkan. More Lebih
recently, Fortemps and Roubens (1996) proposed a method for comparing fuzzy numbers based baru-baru ini, Fortemps dan Roubens (1996) mengusulkan sebuah metode untuk membandingkan bilangan fuzzy berbasis
on the compensation of the areas determined by the membership functions, which has a very terhadap kompensasi daerah ditentukan oleh fungsi keanggotaan, yang memiliki sangat
desirable property that allows us to model the distance between two fuzzy numbers. properti diinginkan yang memungkinkan kita untuk model jarak antara dua nomor fuzzy. This method Metode ini
is also related to the mean value of a fuzzy number as defined in Dubois and Prade (1987) in the juga terkait dengan nilai rata-rata dari bilangan fuzzy seperti yang didefinisikan dalam Dubois dan Prade (1987) dalam
framework of Dempster-Shafer theory. kerangka teori-Shafer Dempster. In the framework of Dempster-Shafer's theory, a fuzzy Dalam rangka-Shafer Dempster teori, sebuah fuzzy
number nomor
§ §
¥ ¥
is considered to define a set of possible probability distributions; hence the mean value dianggap untuk mendefinisikan satu set distribusi probabilitas mungkin, maka nilai rata-rata
of the fuzzy number jumlah fuzzy
§ §
¥ ¥
is the set of mean values calculated according to all these probability adalah himpunan nilai-nilai rata-rata dihitung menurut semua kemungkinan ini
distributions. distribusi.
The lower mean value Nilai rata-rata yang lebih rendah
¥lŠ ¥ ls
and upper mean value dan atas nilai rata
¥ ¥
Š Š
are calculated from the following distri- dihitung dari berikut didistri-
bution functions bution fungsi
ˆ ŠU ¦ SU |
and dan
ˆ
Š Š
¦ |
as follows. sebagai berikut.
ˆ
Š Š
¦ |
798A@ 798A @
! !
©‹& © <&
' '
“” U) “GŒ ¦Ž5 "" U) "Goe | Z5
(10) (10)
7 7
Page 8 Page 8
ˆ Š9 ¦ S9 |
' '
! !
egq{©l& egq {© l &
' '
“” U) “GV ¦Ž5 "" U) "GV | Z5
(11) (11)
¥lŠ ¥ ls
i i
knm KNM
o o
m m
¦lpUrPˆ Š1 ¦ | LpUrP S1 |
(12) (12)
¥ ¥
Š Š
ilknm ilknm
o o
m m
¦lpUrPˆ | LpUrP
Š Š
¦ |
(13) (13)
Thus, the mean value of the fuzzy number Dengan demikian, nilai rata-rata dari bilangan fuzzy
§ §
¥ ¥
is the interval adalah interval
p p
¥lŠ9#(¥ LŠ9 ¥ # (¥
Š Š
f f
. . Now, if we assume that Sekarang, jika kita berasumsi bahwa
all the values in the interval semua nilai dalam interval
p p
¥lŠ9#(¥ LŠ9 ¥ # (¥
Š Š
f f
are equally likely, then the natural choice for a single real sama-sama mungkin, maka pilihan alami untuk satu nyata
number would be the mid-point of this interval. nomor akan menjadi titik tengah interval itu. The area compensation method basically involves Metode kompensasi daerah pada dasarnya melibatkan
computing the following integral: menghitung integral berikut:
€lw§ € § lw
¥' ¥ '
Hd“•”'p Hd "•" 'p
i‰– i ‰ -
— -
¦ |
r r
R R
b¦ b |
t t
R R
˜rAQ ~ RAQ
(14) (14)
Fortemps and Roubens (1996) prove that the area compensation procedure provides a real Fortemps dan Roubens (1996) membuktikan bahwa prosedur kompensasi daerah memberikan nyata
number that is the mid-point of the interval corresponding to the mean value of a fuzzy number, in jumlah itu adalah titik tengah yang sesuai interval untuk nilai rata-rata dari bilangan fuzzy, di
the sense of the Dempster-Shafer theory, where both sides of the possibility distribution induce a arti dari teori-Shafer Dempster, dimana kedua sisi distribusi kemungkinan membujuk
probability measure. mengukur probabilitas. Thus, we have: Dengan demikian, kita memiliki:
€lw§ € § lw
¥' ¥ '
Hd“•”'p Hd "•" 'p
i i
– -
— -
¦ |
r r
R R
b¦ b |
t t
R R
˜rAQ ~ RAQ
Hd“•”'pA ¥—Š b¥ "•" 'pA Hd ¥-Š b ¥
Š Š
(15) (15)
The area-compensation integral in (14) can be used to compare the fuzzy makespans of poten- Daerah-kompensasi integral dalam (14) dapat digunakan untuk membandingkan makespans fuzzy potensial-
tial schedules. TiAl jadwal. If we are interested in minimizing the makespan, a schedule with fuzzy makespan Jika kita tertarik untuk meminimalkan makespan, jadwal dengan makespan fuzzy
§ §
¥ ¥
– -
is preferred over a schedule with fuzzy makespan lebih disukai daripada jadwal dengan makespan fuzzy
§ §
¥š™ ¥ ™ s
if jika
€lw§ € § lw
¥ ¥
– -
G› G>
¨ ¨
€l œ§ L € œ §
¥š™‰ ¥ Š ™ ‰
. .
3.3.2 Probabilistic Case 3.3.2 Kasus Probabilistik
Suppose we wish to compute the expected value of a function of several random variables such Misalkan kita ingin menghitung nilai yang diharapkan dari suatu fungsi dari beberapa variabel acak seperti
as sebagai
y y
ž Ě
¥ ¥
– -
#(¥š™9#g“9“9“g#”¥—Ÿ # (¥ Š ™ 9 # g "9" 9 "g #" ¥-Y
. . The first step is to obtain the joint Langkah pertama adalah untuk mendapatkan sambungan
… †‰ˆ ... † ‰
of the random variables, dari variabel acak,
ie, yaitu,
w ¦ w |
– -
#9“9“9“„#(¦uŸ # 9 "9" 9 ""#(| uy
, which in the case of independent random variables is merely the product of the , Yang dalam kasus variabel acak independen hanyalah produk dari
individual individu
… †‰ˆ ... † ‰
s. s. Then, to obtain the Kemudian, untuk mendapatkan
€‡†‰ˆ € ‡ † ‰
of dari
y y
, ,
ˆ ™ “” ™ ""
, we have to compute a multiple integral , Kita harus menghitung kelipatan integral
over a polytope lebih dari sebuah polytope
… ...
that is parametric in t. yang parametrik dalam t.
ˆ ™ “” ™ ""
iIi iii
p1p1p p1p1p
i9 `¡£¢¥¤ i9 `¡£¢¥¤
¦ |
– -
#9“g“9“g#(¦uŸ– pUr¦¦ # 9 "g" 9 "g # (| uy-Pur | |
– -
p1p1p(r¦¦uŸ p1p1p (r | | uy
…l “” ... L ""
!E„ ! E "
– -
“” –Œ ¦ ""-Å |
– -
ŒW§ ŒW §
– -
“” ‰#g“9“9“g# "" ‰ # g "9" 9 "g #
„ "
Ÿ Y
¦$e¦#(¦ ªd#9“g“9“„#”¦uŸG –Œ “«5 | $ E | # (| ª d # 9 "g" 9 ""#"| uŸG-å "« 5
(16) (16)
Once Setelah
ˆ ™ “” ™ ""
has been obtained, the telah diperoleh,
…‡†‰ˆ ... ‡ † ‰
can be obtained through differentiation and the expected dapat diperoleh melalui diferensiasi dan yang diharapkan
value of nilai
y y
may be computed. dapat dihitung. However, computing the parametric multiple integral in (16) is an Namun, menghitung beberapa parametrik integral dalam (16) adalah
expensive operation and is not tractable, even when the number of random variables is moderately operasi mahal dan tidak penurut, bahkan ketika jumlah variabel acak cukup
large. besar.
8 8
Page 9 Page 9
3.3.3 Comparison 3.3.3 Perbandingan
It is instructive to compare the steps involved in computing the makespan in the fuzzy set case with Ini adalah pelajaran untuk membandingkan langkah-langkah yang terlibat dalam perhitungan makespan dalam kasus himpunan fuzzy dengan
the probabilistic case. kasus probabilistik.
When the processing times are given by fuzzy numbers, the makespan, which is a function Jika waktu proses yang diberikan oleh bilangan fuzzy, makespan, yang merupakan fungsi
of these fuzzy numbers, can be computed very efficiently (although approximately) through inter- angka-angka fuzzy, bisa dihitung sangat efisien (meskipun kurang lebih) melalui inter-
val arithmetic at various val aritmatika di berbagai
Q Q
-levels, as in (4) and (7) and the objective computed through the one- -Level, seperti dalam (4) dan (7) dan tujuan dihitung melalui satu-
dimensional integral in (14). dimensi integral dalam (14). However, in the probabilistic case, the distribution of the makespan Namun, dalam kasus probabilistik, distribusi makespan
is very difficult to obtain, since it has to be obtained through several parametric multiple integrals sangat sulit diperoleh, karena itu harus diperoleh melalui beberapa beberapa integral parametrik
as in (16). seperti di (16).
Furthermore, we note the following interesting properties of the area compensation operator, Selanjutnya, kami mencatat sifat menarik berikut operator kompensasi daerah,
which have their counterparts in the probabilistic approach. yang telah rekan-rekan mereka dalam pendekatan probabilistik. The resemblance of Kemiripan
€l œ§ L € œ §
¥' ¥ '
to the ke
expected value operator operator diharapkan nilai
s s
¥' ¥ '
is indeed striking (see (6) and (9)). memang mencolok (lihat (6) dan (9)).
€l € l
§ §
¥˜ ¥ ~
§ §
i i
€— € -
§ §
¥ˆ $ ¥ $
¨ ¨
€— € -
§ §
i i
(17) (17)
€l € l
‚…„ , ... "
¦ |
p p
§ §
¥b# § ¥ b # §
i i
f¬ V f ¬ V
‚…„ , ... "
¦ |
pƒ¨ pƒ ¨
€l X§ € l § X
¥' y# ¥ 'y #
€l P§ P l € §
i i
†f † f
(18) (18)
4 Overview of Approach 4 Sekilas Pendekatan
The key ideas that we will use to apply the concepts presented in the previous section are as Ide-ide kunci yang akan kita gunakan untuk menerapkan konsep yang disajikan pada bagian sebelumnya adalah sebagai
follows: berikut:
1. Evaluation of a given schedule under uncertainty 1 jadwal. Evaluasi diberikan di bawah ketidakpastian
This step involves the calculation of the start-times and end-times at various Langkah ini melibatkan perhitungan waktu mulai dan akhir-kali di berbagai
Q Q
-levels, through -Tingkat, melalui
a set of recurrence relations. seperangkat hubungan kambuh. These computations typically involve the use of interval arith- Perhitungan ini biasanya melibatkan penggunaan arith interval-
metic. metic.
2. Generalization of Expressions 2. Generalisasi Mengungkapkan
Once the expressions for calculating the start- and end-times for the tasks for a given sched- Setelah ekspresi untuk menghitung-awal dan akhir-kali untuk tugas-tugas yang diberikan sched-
ule (sequence of tasks) have been derived, the next step is to generalize these to account for ule (urutan tugas) telah diperoleh, langkah selanjutnya adalah generalisasi ini untuk memperhitungkan
all possible schedules. semua kemungkinan jadwal. This is done first by introducing binary variables that model either the Hal ini dilakukan terlebih dahulu dengan memperkenalkan model variabel biner yang baik
precedence relationships between the tasks, or the allocation of tasks to specific processing diutamakan hubungan antara tugas, atau alokasi tugas untuk pengolahan tertentu
units. unit.
3. Optimization 3. Optimasi
The area compensation integral will be used to compare the objective functions of potential Kompensasi daerah integral akan digunakan untuk membandingkan fungsi objektif potensi
schedules. jadwal. The optimization models use a discretization approximation to the calculation of Model optimasi menggunakan pendekatan beda terhadap perhitungan
9 9
Page 10 Page 10
the one-dimensional integral in (14). satu-dimensi integral dalam (14).
1 1
Furthermore, the benefits of using a large number of Selain itu, manfaat menggunakan sejumlah besar
discretization points are also less pronounced - with very small improvements in the estima- beda poin juga kurang diucapkan - dengan perbaikan kecil yang sangat dalam memperkirakan the-
tion of the integral and an order of magnitude larger computation time. tion dari integral dan urutan waktu komputasi yang lebih besar besaran. In some cases, we Dalam beberapa kasus, kita
also consider the generation of moves for local search techniques. juga mempertimbangkan generasi bergerak untuk teknik pencarian lokal.
5 The Flowshop Problem 5 Masalah flowshop
Stage 2 Tahap 2
Stage 1 Tahap 1
Stage 3 Tahap 3
Reactor Reaktor
Centrifuge Sentrifuse
Tray Dryer Baki pengering
Product Produk
Figure 3: Gambar 3:
Flowshop Plant Tanaman flowshop
Flowshop plants (see Fig.3) are multiproduct batch plants where the jobs associated with the tanaman flowshop (lihat Gbr.3) merupakan tanaman batch multiproduct mana pekerjaan yang terkait dengan
manufacturing of product orders use the same set of processing units in the same order (Birewar pembuatan produk pesanan menggunakan set yang sama dari unit pengolahan dalam urutan yang sama (Birewar
and Grossmann, 1989). dan Grossmann, 1989). A solution to the flowshop scheduling problem specifies the order in which Sebuah solusi untuk masalah penjadwalan flowshop menentukan urutan yang
jobs are processed in each unit. pekerjaan diproses di unit masing-masing.
Given are Yang diberikan adalah
® ®
products produk
#«¯l#9“9“9“° # «¯ l # 9" 9 "9" ˘
, ,
24± 24 ±
, that are to be manufactured in a flowshop plant with , Yang akan diproduksi di pabrik flowshop dengan
² ²
processing stages tahap pengolahan
³e¦#yªP#9“9“9“‰# E ³ | # y P ª # 9 "9" 9 "‰ #
² ²
, ,
l2¶µ l2 ¶ μ
. . Each product requires processing in all of the Setiap produk memerlukan pengolahan dalam semua
² ²
stages tahap
and follows the same sequence of operations, ie, a permutation flowshop plant. dan mengikuti urutan operasi yang sama, yaitu, sebuah pabrik flowshop permutasi. In the case of Dalam kasus
Unlimited Intermediate Storage (UIS) mode of operation, an unlimited number of batches may be Unlimited Intermediate Storage (UIS) Mode operasi, jumlah yang tidak terbatas mungkin batch
held in storage between stages. diselenggarakan di penyimpanan antara tahap. The computation of the makespan (of a given processing sequence) Perhitungan makespan (dari urutan pemrosesan yang diberikan)
for the UIS flowshop plant involves a series of recurrence relations for the completion times untuk pabrik flowshop UIS melibatkan serangkaian hubungan kambuh waktu penyelesaian
·«“ ³¹ · «" ³ ¹
of product in position produk dalam posisi
@ @
in the sequence in stage . dalam urutan tahap. The relations for the completion times for the Hubungan untuk waktu penyelesaian untuk
deterministic case are given below (Rajagopalan and Karimi, 1989): kasus deterministik diberikan di bawah ini (Rajagopalan dan Karimi, 1989):
·«“ ³¹ · «" ³ ¹
tluEv tluEv
c·«“ ° C «"
o o
–zº -Z º
¹1#«·«“ ¹ 1 #«·«"
º º
¹ ¹
o o
– -
Ž b»d”¹ É b »d" ¹
Y Y
@ @
F e¦#³Yƒ´¼Fje F e | Yƒ '¼ Fje ³ #
(19) (19)
·«“ · «"
– -
¹ ¹
·«“ · «"
–zº -Z º
¹ ¹
o o
– -
b» b »
– -
¹ ¹
Yƒ´½Fje Yƒ '½ Fje
(20) (20)
·y“ · Y "
– -
·«“ · «"
o o
–zºƒ– -Z º ƒ-
a»ƒ¸ sebuah »ƒ ¸
– -
Y Y
@ @
Fje Fje
(21) (21)
‚ ,
7 7
·«“³Ÿ ¾ · «" ³ Y ¾
(22) (22)
1 1
A rectangular approximation of this integral leads to large errors, when the number of discretization points is Sebuah pendekatan persegi panjang ini mengarah integral kesalahan besar, ketika jumlah poin diskritisasi adalah
low; while a piecewise quadratic approximation (Simpson's rule) gives much better results. rendah, sementara pendekatan kuadrat sesepenggal ('s aturan Simpson) memberikan hasil yang lebih baik. In the results presented, Pada hasil yang disajikan,
Simpson's rule was used to approximate the integral in (14). Simpson's aturan digunakan untuk perkiraan integral dalam (14).
10 10
Page 11 Page 11
where mana
»ƒ¸³¹ »Ƒ ¸ ³ ¹
is the processing time of the product in position adalah waktu pengolahan produk pada posisi
@ @
in the sequence in stage and dalam urutan di panggung dan
‚ ,
7 7
is adalah
the makespan of the corresponding schedule. makespan dari jadwal yang sesuai.
For the case where the processing times are represented by fuzzy numbers, the above relations Untuk kasus di mana waktu pengolahan diwakili oleh bilangan fuzzy, di atas hubungan
have to be modified using the extension principle. harus diubah dengan menggunakan prinsip ekstensi. The integral in (14) is approximated by a Integral dalam (14) didekati dengan
summation of the function values at specific values of penjumlahan dari nilai fungsi pada nilai-nilai tertentu
Q Q
. . Thus, the interval [0,1] is discretized to Dengan demikian, interval [0,1] adalah didiskritisasi untuk
points poin
!E„ ! E "
– -
# #
„ "
™9#g“9“9“g# ™ 9 # g "9" 9 "g #
„ "
$5 $ 5
, with stepsize , Dengan stepsize
¤h¤ H ¤ ¤
eU¿˜ eu ¿~
q eU q eu
. .
·«“ ³¹”À · «" ³ ¹ "À
tluEv tluEv
|·y“ | Y · "
o o
–zº -Z º
¹ ¹
º º
À1#«·«“ A1 #«·«"
º º
¹ ¹
o o
–zº -Z º
À‰ Ž a»ƒ¸³¹”À À ‰ é a »ƒ ¸ ³ ¹" À
Y Y
@ @
FÁe¦#³Yƒ´¼FÁe¦#†Y Fae | # ³ Yƒ 'Fae ¼ | # † Y
„ "
(23) (23)
·«“ · «"
– -
¹”À ¹ "à
·«“ · «"
–zº -Z º
¹ ¹
o o
–zº -Z º
Àw b» Aw b »
– -
¹”À ¹ "à
Yƒ´½FÁe¦#³Y Yƒ '½ Fae | # ³ Y
„ "
(24) (24)
·y“ · Y "
– -
À À
·«“ · «"
o o
–zºƒ–zº -Z º ƒ-z º
Àœ a»ƒ¸ Aoe sebuah »ƒ ¸
– -
À À
Y Y
@ @
F e¦#³Y F e | # ³ Y
„ "
(25) (25)
‚ ,
7 7
À À
·«“³Ÿ ¾'À · «" ³ ¾ 'y
Y Y
„ "
(26) (26)
With the above relations for the completion times of the various operations for a specific Dengan hubungan di atas untuk waktu penyelesaian berbagai operasi untuk pesan tertentu
processing sequence, we can formulate an MILP model for selecting the sequence with optimal urutan pengolahan, kita dapat merumuskan model MILP untuk memilih urutan dengan optimal
makespan. makespan. In this MILP model, we introduce binary variables Dalam model MILP, kami memperkenalkan variabel biner
wEÂ Wea
w w
denote the assignment menunjukkan tugas
of product produk
to position ke posisi
@ @
in the processing sequence. dalam urutan pemrosesan. A sequence with fuzzy makespan Sebuah urutan dengan makespan fuzzy
¥ ¥
– -
is adalah
preferred over a sequence with makespan lebih disukai daripada urutan dengan makespan
¥š™ ¥ ™ s
if jika
y y
¥ ¥
– -
4› 4>
y y
¥š™y Y ¥ ™ s
, with , Dengan
y y
¥' ¥ '
calculated using dihitung menggunakan
(14). (14). The MILP model (FSP) is as below: Model MILP (FSP) adalah sebagai berikut:
(FSP) Min (FSP) Min
ÃgÄwÅ ÃgÄwÅ
'¤X¤Ž¿BƦ pP ³eU¿¦ªB pEÇ '¤ X ¤ é ¿Bae | PP ³ eu ¿| ª B PEC
À À
¤X€GÀ–pA c·«“³Ÿ ¾'À X € GA-pA c ¤ · «" ³ Y ¾ 'À
r r
È·«“³Ÿ ¾'À E ° «" ³ ¾ 'y
t t
(27) (27)
·y“ ³¹”À³ÉÊV · Y "³ ¹" à ³ ÉÊV
·«“ · «"
o o
–zº -Z º
¹ ¹
º º
À À
º º
É É
Ç Ç
 Â
» »
 Â
¹”À³Éwp ¹ "à ³ EWP
wd WDA
Y Y
@ @
V©ªP#³Yƒ´¦#³Y V © ª P # ³ Yƒ '| # ³ Y
„ "
#«Ë # «Ë
(28) (28)
·y“ ³¹”À³ÉÊV · Y "³ ¹" à ³ ÉÊV
·«“ · «"
º º
¹ ¹
o o
–zº -Z º
À À
º º
É ÌÇ É IC
 Â
» »
 Â
¹”À³Éwp ¹ "à ³ EWP
wd WDA
Y Y
@ @
#³Yƒl # ³ Yƒl
„ "
#«Ë # «Ë
(29) (29)
·«“ · «"
–z– -Z-
À”É À "É
Ç Ç
 Â
» »
 Â
– -
À³Éœp À ³ Éœp
wEÂ Wea
– -
Y Y
„ "
#%Ë #% Ë
(30) (30)
·y“ · Y "
– -
¹³À”É ³ ¹ à "É
·«“ · «"
–zº -Z º
¹ ¹
o o
–zº -Z º
À À
º º
ÉŽ ÌÇ EZ IC
 Â
» »
 Â
¹”À³Éwp ¹ "à ³ EWP
wd WDA
– -
Yƒ´—V©ªPͳY Yƒ'-V © ª Pi ³ Y
„ "
#«Ë # «Ë
(31) (31)
·«“ · «"
– -
À”É À "É
·«“ · «"
o o
–zºƒ–zº -Z º ƒ-z º
À À
º º
É$ É $
Ç Ç
 Â
» »
 Â
– -
À³Éœp À ³ Éœp
wd WDA
Y Y
@ @
V©ªPͳY V © ª Pi ³ Y
„ "
#«Ë # «Ë
(32) (32)
Ç Ç
w w
e e
Y Y
(33) (33)
Ç Ç
 Â
wd WDA
e e
Y Y
@ @
(34) (34)
·y“ ³¹”À³ÉÊV · Y "³ ¹" à ³ ÉÊV
H H
Y Y
@ @
#z¦# # Z | #
„ "
#«Ë # «Ë
(35) (35)
wd WDA
2 2
! !
Hƒ#9ed5 Hƒ # 9ed5
Y Y
# #
@ @
(36) (36)
where the subscripts dimana subskrip
Ë Ë
indicate the left and right end-points used in the interval-arithmetic; menunjukkan ujung kiri dan kanan-poin yang digunakan dalam aritmatika interval;
thus, demikian,
ˇ2 Ë ‡ 2
s s
Î!EÏ Î! Ei
#(ÐP5 # (DP5
. . In (FSP), (27) represents the discrete approximation of the area-compensation Dalam (FSP), (27) merupakan pendekatan diskrit kompensasi area
integral applied to the makespan, with the terpisahkan diterapkan makespan, dengan
¤œ€ ¤ € œ
s denoting the Simpson coefficients used in the ap- s Simpson yang menunjukkan koefisien yang digunakan dalam pendekatan yang-
proximation of the integral. proximation dari integral. Constraints (28)-(32) are the generalization of the completion time Kendala (28) - (32) adalah generalisasi dari waktu penyelesaian
11 11
Page 12 Page 12
constraints in (23)-(25) for any sequence. kendala dalam (23) - (25) untuk urutan apapun. Constraints (33) and (34) are the well-known assign- Kendala (33) dan (34) adalah terkenal menetapkan-
ment constraints that specify that each product kendala pemerintah yang menentukan bahwa setiap produk
should be assigned to only 1 position, and that harus diserahkan kepada hanya 1 posisi, dan bahwa
each position setiap posisi
@ @
should be assigned to only one product. harus diserahkan kepada hanya satu produk. Model (FSP) can also be extended to the Model (FSP) juga dapat diperpanjang dengan
case where there are set-up times between the processing of different products. kasus di mana ada set-up kali antara pengolahan produk yang berbeda. These set-up times Ini set-up kali
can either be sequence dependent or unit dependent. bisa baik menjadi urutan tergantung atau unit tergantung.
5.1 Illustrative Example Contoh Ilustrasi 5.1
Table 1: TFN Descriptions for 5-Product, 4-Stage Example Tabel 1: Deskripsi TFN untuk 5-Produk, 4-Tahap Contoh
Product Produk
Stage 1 Tahap 1
Stage 2 Tahap 2
Stage 3 Tahap 3
Stage 4 Tahap 4
1 1
(16.083, 17, 18.405) (41.680, 44, 45.036) (31.369, 32, 34.011) (21.505, 22, 22.205) (16,083, 17, 18,405) (41,680, 44, 45,036) (31,369, 32, 34,011) (21,505, 22, 22,205)
2 2
(20.056, 22, 23.490) (16.632, 19, 19.692) (23.169, 24, 27.099) (39.606, 44, 48.935) (20,056, 22, 23,490) (16,632, 19, 19,692) (23,169, 24, 27,099) (39,606, 44, 48,935)
3 3
(11.487, 13, 15.002) (28.634, 30, 31,787) (49.171, 50, 56,611) (30,513, 33, 37.988) (11,487, 13, 15,002) (28,634, 30, 31787) (49,171, 50, 56.611) (30.513, 33, 37,988)
4 4
(48.799, 50, 56,274) (34.765, 40, 42,271) (14.403, 15, 15,259) (34.457, 36, 36.738) (48,799, 50, 56274) (34,765, 40, 42271) (14,403, 15, 15259) (34,457, 36, 36,738)
5 5
(14.575, 16, 18.052) (17.832, 20, 22.181) (33.513, 37, 37.233) (25.122, 27, 31.279) (14,575, 16, 18,052) (17,832, 20, 22,181) (33,513, 37, 37,233) (25,122, 27, 31,279)
220 220
230 230
240 240
250 250
260 260
270 270
280 280
290 290
0 0
0.1 0.1
0.2 0.2
0.3 0.3
0.4 0.4
0.5 0.5
0.6 0.6
0.7 0.7
0.8 0.8
0.9 0.9
1 1
Membership Function of Makespans of Different Sequences Keanggotaan Fungsi Makespans dari Urutan Berbeda
Time Waktu
Membership Keanggotaan
Opt.Seq. Opt.Seq. 5−2−3−1−4 5-2-3-1-4
Seq. Seq. 5−2−3−4−1 5-2-3-4-1
Seq. Seq. 2−1−3−4−5 2-1-3-4-5
Figure 4: Gambar 4:
Membership Functions of Makespans Fungsi Keanggotaan Makespans
12 12
Page 13 Page 13
Figure 4 presents the membership functions of the makespans of three processing sequences Gambar 4 menyajikan fungsi keanggotaan dari makespans tiga urutan pengolahan
for a 5-product, 4-stage example. untuk produk 5, 4-tahap contoh. These membership functions are obtained by solving the FSP Fungsi-fungsi keanggotaan diperoleh dengan memecahkan FSP
model and looking up the model dan melihat ke atas
·y“ · Y "
values at the various nilai pada berbagai
Q Q
-levels. -Tingkat. The data for this example can be found Data untuk contoh ini dapat ditemukan
in Table 1. pada Tabel 1. The processing times are specified as triplets corresponding to a Triangular Fuzzy Lamanya proses ditetapkan sebagai kembar tiga yang berhubungan dengan Fuzzy segitiga
Number (TFN) description. Number (TFN) deskripsi. The computations were performed with a 21-point discretization. Perhitungan dilakukan dengan titik diskritisasi 21. The The
CPU time for solving the FSP model was under 5 secs. CPU waktu untuk menyelesaikan model FSP di bawah 5 detik. Table 2 summarizes the key results from Tabel 2 merangkum hasil kunci dari
this example. contoh ini.
Table 2: Results for the Example Problem Tabel 2: Hasil untuk Contoh Soal
Makespan (time units) Makespan (unit waktu)
Sequence Urutan
Objective value (AC) Optimistic Most Likely Pessimistic Tujuan Nilai (AC) Kemungkinan Pesimistis Paling Optimis
5-2-3-1-4 5-2-3-1-4
239.809 239.809
225.59 225.59
238 238
258.108 258.108
5-2-3-4-1 5-2-3-4-1
239.967 239.967
224.734 224.734
239 239
258.108 258.108
2-1-3-4-5 2-1-3-4-5
264.961 264.961
249 249
263 263
284.845 284.845
The optimal sequence ( Urutan yang optimal (
€ €
value of 239.809 units) was determined to be '5-2-3-1-4' with a most nilai 239,809 unit) bertekad untuk menjadi '5 -2-3-1-4 'dengan yang paling
likely makespan value of 238 units, an optimistic makespan of 225.59 units and a pessimistic makespan kemungkinan nilai 238 unit, makespan optimis 225,59 unit dan pesimis
makespan of 258.108 units. makespan dari 258,108 unit. This information is again obtained by simply looking up the values Informasi ini lagi diperoleh dengan hanya mencari nilai-nilai
of the variables dari variabel-variabel
·«“†ÑÓÒ† · «" † † Noo
r r
, ,
·«“†ÑÓÒ · «" † Noo
– -
r r
and dan
·«“†ÑÓÒ · «" † Noo
– -
t t
. . Also shown in Figure 4 is the membership function of Juga ditunjukkan pada Gambar 4 adalah fungsi keanggotaan
the makespan of sequence '5-2-3-4-1': note that although this sequence has a better optimistic makespan urutan -2-3-4-1 '5 ': dicatat bahwa meskipun urutan ini memiliki lebih optimis
makespan than sequence '5-2-3-1-4', it is not optimal with respect to the makespan dari urutan -2-3-1-4 '5 ', tetapi tidak optimal berkaitan dengan
€ €
of the makespan, dari makespan,
with a marginally higher dengan sedikit lebih tinggi
€ €
value. nilai. Indeed, if we change the objective function so that we want Memang, jika kita mengubah fungsi objektif sehingga yang kita inginkan
a sequence for optimal optimistic makespan, we get sequence '5-2-3-4-1' as optimal. urutan untuk makespan optimis optimal, kami mendapatkan urutan -2-3-4-1 '5 'sebagai optimal. If we wish Jika kita ingin
to find the sequence with minimum pessimistic makespan, we obtain either of the above two se- untuk menemukan urutan dengan makespan minimum pesimis, kita mendapatkan salah satu dari dua di atas se-
quences since both of them have the minimum pessimistic makespan of 258.108 units. quences karena keduanya memiliki makespan pesimis minimum 258,108 unit. Finally, Akhirnya,
sequence '2-1-3-4-5' is sub-optimal with respect to all measures - Urutan -1-3-4-5 '2 'adalah sub-optimal terhadap semua langkah -
€ €
value, optimistic, most nilai, optimis, sebagian besar
likely and pessimistic makespans. kemungkinan dan pesimis makespans. Clearly, sequences '5-2-3-1-4' and '5-2-3-4-1' are preferrable Jelas, urutan -2-3-1-4 '5 'dan '5 -2-3-4-1 adalah preferrable
over sequence '2-1-3-4-5' on two counts: 1) in terms of the atas urutan -1-3-4-5 '2 'pada dua hal: 1) dalam hal
€ €
values, and 2) in terms of the nilai, dan 2) dalam hal
spread of the makespan. penyebaran makespan. While the makespan of sequence '5-2-3-1-4' has a spread of 32.518 units Sedangkan makespan urutan -2-3-1-4 '5 'memiliki penyebaran 32,518 unit
(ie, 258.108-225.59), sequence '2-1-3-4-5' has a spread of 35.845 units. (Yaitu, 258,108-225,59), urutan -1-3-4-5 '2 'memiliki penyebaran 35,845 unit.
5.2 Computational Results 5.2 Komputasi Hasil
In the following examples, all MILP models were formulated using GAMS and solved using Dalam contoh berikut, semua model MILP dirumuskan menggunakan GAMS dan diselesaikan dengan menggunakan
CPLEX 7.0 on a Pentium III/930 MHz machine running Linux. CPLEX 7,0 pada mesin Pentium III/930 MHz menjalankan Linux. Table 3 displays the solution Tabel 3 menampilkan solusi
13 13
Page 14 Page 14
times required for model (FSP) when applied to problems of differing sizes. waktu yang diperlukan untuk model (FSP) ketika diterapkan untuk masalah ukuran yang berbeda. In these examples, Pada contoh ini,
the durations were assumed to be given by asymmetric Triangular Fuzzy Numbers (TFNs). durasi diasumsikan diberikan oleh asimetris Segitiga Fuzzy Number (TFNs). The The
results are from several examples, with the TFNs randomly generated. hasil dari beberapa contoh, dengan TFNs secara acak. In these models, a 21-point Dalam model ini, sebuah 21-point
discretization was used for evaluating the integral in (14). diskritisasi digunakan untuk mengevaluasi integral dalam (14). As can be seen, the computational times Seperti dapat dilihat, maka komputasi kali
required are quite small. yang diperlukan cukup kecil. In the worst case about half an hour was required for one of the 12- Dalam kasus terburuk sekitar setengah jam diperlukan untuk satu dari 12 -
product, 4-stage problems. produk, 4-tahap masalah. Note that the 12-product examples contain as many as 48 tasks with Perhatikan bahwa 12-produk contoh berisi sebanyak 48 tugas dengan
uncertain durations, which would pose a formidable dimensionality problem to a probabilistic jangka waktu yang tidak pasti, yang akan menimbulkan masalah dimensi tangguh ke probabilistik
model. model.
Table 3: Computational Times for Flowshop Problems Tabel 3: Times Komputasi untuk Masalah flowshop
CPU Time (secs) CPU Time (secs)
Products Stages Produk Tahapan
Min. Min. Mean Max. Mean Max.
5 5
4 4
4 4
4.5 4.5
6 6
4 4
27 27
101 101
200 200
8 8
5 5
160 160
178 178
237 237
3 3
8 8
31 31
122 122
12 12
4 4
296 296
788 788
1598 1598
5.3 Remarks - Local Search Techniques 5.3 Keterangan - Teknik Pencarian Lokal
For larger problems where the number of products to be processed is quite large, say 20-50 prod- Untuk masalah yang lebih besar dimana jumlah produk yang akan diolah cukup besar, misalnya 20-50 prod-
ucts, solving model (FSP) to optimality can be prohibitive. ucts, pemecahan model (FSP) untuk optimalitas dapat menjadi penghalang. In these cases, the use of a local search Dalam kasus ini, penggunaan pencarian lokal
algorithm such as Simulated Annealing (Kirkpatrick et al. , 1983; Aarts and Korst, 1989) or Tabu Algoritma seperti Simulated Annealing (et al Kirkpatrick;., 1983 Aarts dan Korst, 1989) atau Tabu
Search (Glover, 1990) may be preferrable, since these algorithms can obtain good quality solutions Pencarian (Glover, 1990) mungkin preferrable, karena algoritma ini dapat memperoleh kualitas solusi yang baik
within reasonable time. dalam waktu yang wajar. However, these algorithms also have significant drawbacks - they do not Namun, algoritma ini juga memiliki kelemahan yang signifikan - mereka tidak
provide any guarantee on the quality of the solution obtained, and it is often impossible to tell how memberikan jaminan atas kualitas dari solusi yang diperoleh, dan sering tidak mungkin untuk menceritakan bagaimana
far the current solution is from optimality. Sejauh ini, solusi saat ini dari optimal.
In this section, we outline results from a Tabu Search implementation for flowshop optimiza- Pada bagian ini, kita garis besar hasil dari implementasi Tabu Search untuk optimiza flowshop-
tion. tion. We chose Tabu Search for the advantages that it offers over Simulated Annealing and Genetic Kami memilih Tabu Search untuk keuntungan yang menawarkan lebih dari Simulated Annealing dan Genetik
Algorithms. Algoritma. Tabu Search is the most deterministic of the three techniques and also has fewer tun- Tabu Search adalah yang paling deterministik dari tiga teknik dan juga memiliki sedikit tun-
able parameters. parameter mampu. We implemented a variant of Tabu Search called Reactive Tabu Search (RTS) Kami menerapkan varian dari Tabu Search disebut Reactive Tabu Search (RTS)
(Battiti and Tecchiolli, 1994; Schmidt, 1998) which has the desirable property that the length of (Battiti dan Tecchiolli, 1994; Schmidt, 1998) yang memiliki properti yang diinginkan bahwa panjang
the tabu list is dynamically adjusted during the progress of the algorithm. daftar tabu secara dinamis disesuaikan selama kemajuan algoritma. The dynamic adjust- Dinamika menyesuaikan-
ment of the tabu list helps to automatically intensify the search in promising regions or diversify pemerintah dari daftar tabu untuk secara otomatis membantu mengintensifkan pencarian di wilayah menjanjikan atau diversifikasi
14 14
Page 15 Page 15
the search in unexplored regions. pencarian di daerah yang belum dijelajahi. RTS stores all solutions (through the use of hashtables, digital RTS menyimpan semua solusi (melalui penggunaan hashtables, digital
search trees etc.) as they are visited during its progress, so that it is possible to recognize whether cari pohon, dll) karena mereka kunjungi selama perkembangannya, sehingga memungkinkan untuk mengenali apakah
a particular solution has been encountered before. solusi tertentu telah ditemukan sebelumnya. The frequency with which solutions reappear Frekuensi dengan solusi yang muncul kembali
indicates whether the search should be intensified or diversified. menunjukkan apakah pencarian harus ditingkatkan atau diversifikasi. RTS also assumes that if a num- RTS juga mengasumsikan bahwa jika seorang num-
ber of solutions have been encountered several times, the search has been trapped in a basin of ber solusi telah ditemukan beberapa kali, cari telah terperangkap dalam cekungan
attraction. daya tarik. Consequently, a number of random moves are made to escape from the current region. Akibatnya, sejumlah langkah acak dibuat untuk melarikan diri dari daerah saat ini.
There are a few tunable parameters used by the RTS algorithm that relate to the threshold for a Ada beberapa parameter yang merdu yang digunakan oleh algoritma RTS yang berhubungan dengan ambang batas untuk sebuah
”frequently seen” solution, the number of frequently seen solutions before performing the escape "Sering terlihat" solusi, jumlah terlihat solusi sering sebelum melakukan melarikan diri dari
sequence, the factors by which the tabu list size is altered etc. The reader is referred to Battiti and urutan, faktor-faktor di mana ukuran daftar iterasi tabu diubah dll Pembaca dimaksud Battiti dan
Tecchiolli (1994) for a more detailed description of the RTS algorithm. Tecchiolli (1994) untuk keterangan lebih rinci dari algoritma RTS.
For the flowshop problem, a permutation string of the products is a natural characterization Untuk masalah flowshop, string permutasi dari produk adalah karakterisasi alami
of the solution. dari solusi. A move was defined to be any pair-wise interchange of the products in the permu- Sebuah langkah didefinisikan akan ada-bijaksana pertukaran sepasang produk di permu the-
tation. tasi. Thus the neighborhood of allowable moves from a solution is of size Jadi sekitar bergerak diijinkan dari solusi adalah ukuran
Ô— c® O-c ®
™ ™
. . Furthermore, Selanjutnya,
the computation of the makespan of a given solution can be done in perhitungan dari makespan dari solusi yang diberikan dapat dilakukan di
Ô‰ c®Õp O ‰ c ® op
² ²
p p
, where is , Dimana
the number of jumlah
Q Q
-levels at which the calculation is performed. -Tingkat di mana perhitungan dilakukan. The worst case complexity of one Kompleksitas kasus terburuk dari satu
RTS iteration is determined by the evaluation of the makespans (an RTS iterasi ditentukan oleh evaluasi makespans (sebuah
Ô‰ c®Öp O ‰ c ® op
² ²
p p
operation) of operasi) dari
all the semua
Ô‰ c® O ‰ c ®
™ ™
neighbors. tetangga. Thus, the complexity of one RTS iteration is Dengan demikian, kompleksitas dari satu iterasi RTS
Ô‰ c®Ø×p O C ‰ ® Ø × p
² ²
p p
. . Figure Gambar
5 shows the CPU time required for 1 RTS iteration as a function of the problem size (number of 5 menunjukkan waktu yang dibutuhkan untuk CPU 1 iterasi RTS sebagai fungsi dari ukuran masalah (jumlah
products) and the number of produk) dan jumlah
Q Q
-levels at which the computation is performed. -Tingkat di mana perhitungan dilakukan. The CPU time for Pada saat CPU untuk
one move when the makespan evaluations are carried out at 101 satu bergerak saat evaluasi makespan dilakukan di 101
Q Q
-levels for 3-stage problems is -Tingkat untuk tahap masalah 3 adalah
given by (37), where diberikan oleh (37), dimana
® ®
denotes the number of products. menunjukkan jumlah produk. This relation was obtained by regression. Hubungan ini diperoleh dengan regresi.
The CPU time reported is for a Java (Arnold and Gosling, 1998) implementation on a Pentium III Waktu CPU yang dilaporkan adalah untuk Java (Arnold dan Gosling, 1998) implementasi pada Pentium III
machine running Linux. Mesin yang menjalankan Linux.
“ "
– -
— -
– -
Æd“£ÙBÚÜÛepÜe9H AED "£ ÙBÚÜÛepÜe9H
o o
Ò O
® ®
× ×
qbÙƒ“£”BÙ¦Ù'pÜe9H qbÙƒ "£" BU | Ù'pÜe9H
o o
× ×
® ®
™ ™
ÈHd“£H¦ÝBƃeU”B®q EHD "£ H | ÝBƃeU" B ® q
(37) (37)
Figure 6 displays the initial progress of the RTS for a 20-product, 4-stage problem with sequence- Gambar 6 menampilkan kemajuan awal dari RTS untuk produk 20, 4-tahap masalah dengan urutan-
dependent set-up times on all stages. tergantung set-up kali pada semua tahap. The evaluation of a solution was performed with a 21-point Evaluasi solusi dilakukan dengan 21-point
discretization. diskritisasi. In the initial stages, the algorithm behaves much like a steepest descent heuristic Pada tahap awal, algoritma berperilaku seperti keturunan tajam heuristik
where there is a rapid decrease in the objective function. dimana terjadi penurunan cepat dalam fungsi objektif. Surprisingly enough, even in this early Cukup mengejutkan, bahkan di awal
stage of the algorithm, the escape sequences (sequences of random moves which seek to escape tahap dari algoritma tersebut, escape sequence (urutan bergerak acak yang berusaha untuk melarikan diri
the region of a complex attractor and take the search into a new, unexplored region) are performed, daerah dari attractor kompleks dan mengambil pencarian menjadi belum dijelajahi, wilayah baru) dilakukan,
as indicated by the spikes in Figure 6. The search then proceeds to locate the local minimum
in the new search area. Figure 7 displays the long-term progress of the RTS over the course of
5000 iterations. One can clearly observe the two different time-scales in the search. The solution Solusinya
varies rapidly over a short time horizon, and the escape sequences move the search into unexplored
15 15
Page 16 Page 16
0 0
1 1
2 2
3 3
4 4
5 5
6 6
5 5
10 10
15 15
20 20
25 25
30 30
CPU Time (secs)
ß
Number of Tasks
CPU Time for 1 RTS Iteration as a Function of Problem Size
101 Alpha-levels
51 Alpha-levels
21 Alpha-levels
Figure 5: Gambar 5:
RTS Iteration Time
regions over the longer time horizons. The search was terminated at the end of 5000 iterations and
the best solution was found at iteration 863, with an objective value of 303.005. The total CPU
time required for this problem was around 1400 secs, thus averaging about 0.28 secs per RTS
iteration.
An indication of the quality of the solution can be obtained by solving the corresponding
FSP model to optimality. However, when we attempted to solve the 21-point FSP model for this
particular problem with GAMS/CPLEX, we could not even find an integer feasible solution in
1400 seconds. The LP relaxation of the objective function for this problem was 280.3036, which
indicates that the RTS has done quite well, coming within 8% of the best possible solution in the
same time (1400 secs). The arbitrary termination criteria for the RTS implies that the CPU time
can be adjusted as desired. Furthermore, even after 50,000 CPU secs, CPLEX could not solve
the 21-point FSP model to optimality - the best solution obtained was 295.3863 and the gap was
4.5% (the best node had a value of 282.0548). Thus, we can note that solving these MILP models
to optimality can be computationally expensive. Consequently, the use of local search algorithms
such as RTS that yield near-optimal solutions in significantly smaller computational times is clearly
advantageous. menguntungkan.
16 16
Page 17 Page 17
300 300
310 310
320 320
330 330
340 340
350 350
360 360
370 370
380 380
390 390
50 50
100 100
150 150
200 200
250 250
Objective Function
Iteration
Solutions Found Over RTS Progress
Current Solution
Best Solution so far
Figure 6:
Initial Progress of RTS
300 300
310 310
320 320
330 330
340 340
350 350
360 360
370 370
380 380
390 390
0 0
500 500
1000 1000
1500 1500
2000 2000
2500 2500
3000 3000
3500 3500
4000 4000
4500 4500
5000 5000
Objective Function
Iteration
Solutions Found Over RTS Progress
Current Solution
Best Solution so far
Figure 7:
Long-term Progress of RTS
6 Multistage Flowshop with Parallel Units
6.1 MILP Model
In this section, we address the problem of optimizing the scheduling of multistage flowshop plants
which have parallel units in several stages (see Fig. 8) - an extension of two classical scheduling
17 17
Page 18 Page 18
Stage 1 Stage 2 Stage 3
Orders Pesanan
Figure 8:
Multistage Flowshop with Parallel Units
problems (flowshop and parallel machines problems). Given a production horizon with due dates
for the demands of several products, the key decisions that are to be made are the assignment of
the products to the units as well as the sequencing of products that are assigned to the same unit.
Each product is to be processed only once by exactly one unit of every stage that it goes through.
The objective is usually minimizing the makespan or the total lateness of the orders. We present a
MILP model for obtaining the schedule with minimum total lateness, when the processing times
of the orders are given by fuzzy numbers. The assumptions involved in the model are:
1. 1. The processing times of the various products in the units are given by fuzzy numbers.
2. 2. Due dates are deterministic.
3. 3. Transition times are deterministic and only equipment dependent.
4. 4. There is unlimited intermediate storage between stages.
Assumption 1 seems reasonable, especially when only estimates of the processing times can
be obtained, for instance, when new products are being processed for the first time. Relaxing
assumption 2 will change the nature of the problem. Transition times can also be made fuzzy,
without much problem - however, introducing sequence dependent transitions will considerably
complicate the model.
Much work has been done on deterministic versions of this problem of scheduling flowshop
plants with parallel units. Pinto and Grossmann (1995) presented a continuous-time MILP formu-
lation for minimizing the total weighted earliness and tardiness of the orders. This formulation
was a slot-based model which allocated units and orders on two parallel time coordinates and
matched them with tetra-indexed (stage-order-slot-unit) variables. A later paper by the same au-
thors (Pinto and Grossmann, 1996) incorporated pre-ordering constraints in the model to replace
the tetra-indexed variables by tri-indexed variables (order-stage-unit) and thus reduce the com-
putational effort required to solve the model. McDonald and Karimi (1997) developed an MILP
formulation for the short-term scheduling of single-stage multi-product batch plants with parallel
semi-continuous processing units. More recently, Hui et al. (2000) presented an MILP formulation
which uses tri-indexed variables (order-order-stage) to minimize the total earliness and tardiness of
the orders. With the elimination of the unit index, the formulation required fewer binary variables
18 18
Page 19 Page 19
and reduced computational time.
For our problem where the processing times are given by fuzzy numbers, we present an MILP
model which is a generalizations of the tri-indexed model (Hui et al. , 2000). We also formulated
an MILP model (M1) using the slot-based approach of Pinto and Grossmann (1995), but due to
space restrictions, we restrict our presentation to the tri-indexed model (M2). As before, the timing
computations are performed for different
Q Q
-levels and the objective is a discrete approximation of
the area compensation integral applied to the total lateness.
6.2 Tri-indexed model - (M2)
Given are products (orders)
2D±
and processing stages
Ï Aku
23à
which have processing units
—26µ¦á
. .
Each product
has to be processed on stages
à à
 Â
, and can be processed only on units
µ μ
 Â
. . The The
parameters of interest are the processing times
» »
 Â
¹³À”É
, transition times
¤ â`¹
and due dates
† †
 Â
. .
The model presented in Hui et al. (2000) uses tri-indexed binary variables
¦ |
Âäã
º º
Âæå
º º
á á
to indicate if
product produk
– -
is processed before product
™ ™
in stage
Ï Aku
. . Binaries Binari
7 7
 Â
¹
are used to represent the assignment
of product
to the first processing position in unit , while continuous variables
ç¨è
 Â
¹
are used to
represent assignment of product
to unit . Here we present the model for makespan minimization.
Continuous variables
“ "
7 7
 Â
áéÀ³É
and dan
“³Ë
 Â
áéÀ³É
respectively denote the start and end times of product
in di
stage tahap
Ï Aku
at di
Q Q
-level
„ "
. .
(M2)
tlêìë
y y
Ä Å
'¤X¤Ž¿BÆÜ CfGHƒ“£”0fíÇ
À À
¤œ€GÀhfe
‚
7 7
À À
r r
‚
7 7
À À
t t
(38) (38)
st st
ç
è è
 Â
¹
Ç Ç
¹†îðï
¡æñ”òäódñ”ôæ¤
ç
è è
 Â
î ¹
î î
a¦
•Â
î ᘌ
ª ª
Y Y
# #
è #
öõ W
è #zš26µ
ƒ÷
µ μ
è è
 Â
÷ ÷
暠 #
Ï Aku
2Dà
(39)
Ç Ç
 Â
î¥ïgø
ô o
º º
 Â
î¬ùú
 Â
¦ |
•Â
î ᘌ
e e
Y Y
2D±s#
Ï Aku
23à
(40) (40)
Ç Ç
 Â
î ï9ø
ô o
º º
 Â
î ùú
 Â
¦ |
•Â
î áA
Ç Ç
¹yï
¡ðñ”òðódñ”ôæ¤
7 7
 Â
¹
e e
Y Y
2D±s#
Ï Aku
23à
(41)
ç
è è
 Â
¹
V V
7 7
 Â
¹
Y Y
2D±s#zl26µ
 Â
(42)
Ç Ç
 Â
ï9ø
7 7
 Â
¹
Œ Å
e e
Yƒ´—26µ
 Â
(43) (43)
Ç Ç
¹«ï
¡æñ”ò¥ódñ”ô•¤
ç
 Â
¹
e e
Y Y
2D±s#
Ï Aku
23à
(44) (44)
“ "
7 7
 Â
á°À³É$
Ç Ç
¹«ï
¡æñ”ò¥ódñ”ôæ¤
ç
è è
 Â
¹
pP »
 Â
¹”À³É$ Ȥ–â`¹„ Œ
“ "
7 7
 Â
á î À”É
Y Y
Ï Aku
è #
Ï Aku
2Dàg#
Ï Aku
è è
F F
Ï Aku
# #
2D±s#
„ "
#«Ë
(45)
“ "
7 7
 Â
á°À³É$
Ç Ç
¹«ï
¡æñ”ò¥ódñ
ô ¤
ç
è è
 Â
¹
pP »
 Â
¹”À³É$ Ȥ–â`¹„ Œ
³egq{¦
•Â
î£áä pEâÈ b“
7 7
 Â
î£á°À³É
Y Y
# #
è è
24±u#
¨õ W
è #
Ï Aku
23àí#
„ "
#«Ë
(46) (46)
“ "
7 7
 Â
á á
ò ò
À³É
Ç Ç
¹«ï
¡æñ”ò¥ódñ”ôæ¤
ç
è è
 Â
¹
pP »
 Â
¹”À³É$ Ȥ–â`¹„ Œ
‚
7 7
À³É
Y Y
24±u#
„ "
#%Ë
(47)
19 19
Page 20 Page 20
ç
è è
 Â
¹
V V
H H
Y Y
#z—2{µ
 Â
(48)
ç
è è
 Â
¹
Œ Å
e e
Y Y
#z—2{µ
 Â
(49) (49)
¦ |
•Â
î áû2
! !
Hƒ#9eB5
Y Y
# #
è è
24±u#
¨õ W
è #
Ï Aku
2ˆà
 Â
(50)
7 7
 Â
¹
2 2
! !
Hƒ#9eB5
Y Y
24±u#Ól
 Â
(51) (51)
“ "
7 7
 Â
á°À³É
V V
H H
Y Y
# #
Ï Aku
# #
„ "
#«Ë
(52) (52)
(53)
Constraints (39) specify that if orders
and dan
è è
are consecutive orders and order
is assigned
to unit , then order
è è
is also assigned to unit . Constraints (40) and (41) ensure that each order
has at most one successor and one predecessor respectively in each stage of processing, while
(42) specifies the relation between the assignment and
7 7
variable. variabel. Note that with constraints (39)
and (42), the assignment variables can be relaxed as continuous variables with 0 and 1 as lower
and upper bounds (for proof, please refer to Hui et al. , 2000). Constraints (43) state that each
unit processes at most one starting order and (44) state that every order has to be processed by
a unique unit in each stage. Constraints (45), (46) and (47) govern the relationships between the
starting times of an order in successive stages, the starting times of successive orders in the same
stage and the computation of the makespan respectively. The objective (38) is to minimize the area
compensation of the makespan - this ensures that the makespan of the schedule is the largest end
time of all orders on the final stage.
6.3 Computational Results
Unlike the case of the flowshop problem discussed in Section 5, the MILP models for the multi-
stage flowshop with parallel units do not exhibit good computational performance as the problem
size increases (see Table 4). This is primarily due to the large number of binary variables that are
required for modeling this problem. Although the number of binary variables can be reduced by
taking in to account the forbidden assignments and postulating an appropriate number of slots for
the various units, it increases dramatically with problem size as can be seen from Table 4. The The
solution times reported are for makespan minimization.
Model (M2) was a better formulation than (M1), as can be seen from the reduced computa-
tional times required for the solution (except for the 10-product, 15-units problem). Both models
had the same LP relaxation. Solution of model (M1) for the 10- and 12-product problems was
terminated after 50,000 CPU secs. Model (M2) for the 10-product (25 units) problem was solved
in about 4000 CPU secs, whereas for model (M1), the optimality gap was 11.7% even after 50,000
CPU secs. For the 12-product problem, both models could not be solved to optimality, although for
20 20
Page 21 Page 21
(M2) the optimality gap was smaller than for (M1) after 50,000 CPU secs. The optimality gap for
the 10- and 12-product problems with 25 units is smaller than that for the 10-product problem with
15 units mainly because the 25-unit problems had many assignments that were forbidden, resulting
in more structure for the solution. Even then, these MILP models require excessive computational
resources, as can be seen from the CPU times.
Table 4: Model Characteristics for Parallel Flowshop Problems
Binaries Binari
CPU secs
Optimality Gap (%)
NL-(Units in Stgs)
M1 M1
M2 M2
M1 M1
M2 M2
M1 M1
M2 M2
5-5-(6,3,10,3,3)
241 225
15 15
18 18
0 0
0 0
10-5-(6,3,10,3,3)
721 700
F F
50,000 50,000
4086
11.7 11.7
0 0
10-5-(3,2,3,4,3)
730 700
F F
50,000 50,000
F F
50,000 53.5
56.95
12-5-(6,3,10,3,3)
1000 960
F F
50,000 50,000
F F
50,000 14.5
6.5 6.5
The lateness minimization MILP models also exhibit similar computational performances.
Recall that, for the earliness minimization problem, Pinto and Grossmann (1995) developed a
decomposition strategy which determines feasible assignments that minimize in-process time and
then further determines a minimum-earliness schedule that eliminates unneccesarry set-ups. They Mereka
also used pre-ordering constraints to obtain near-optimal solutions. However, an extension of
these ideas to the case of uncertain processing times is not straightforward. Thus, a local search
technique such as the RTS is particularly relevant for solving larger instances of these problems.
6.4 RTS Implementation
In this section, we discuss some the issues that are critical to the success of the RTS implementation
for this problem.
6.4.1 Solution Characterization and Evaluation
A solution for the multistage flowshop with parallel units is characterized by a feasible allocation
and sequencing of all the products to different processing units in all the stages. One possible Satu kemungkinan
representation of a solution is a set of
ü ü
áðï
r r
)ܵ
r r
) )
strings, with each string denoting the sequencing
of the products allocated to the unit. Of course, the allocation of the products to the various units
must satisfy the constraints that forbid certain allocations as well as (39)-(44). Figure 9 illustrates
one possible solution for a problem where 4 products are to be processed in 2 stages, where the
21 21
Page 22 Page 22
first stage has 2 units in parallel and the second stage has only 1 unit. Thus, product A is processed
first on unit 1 in stage 1 and is followed by B, while C and D are processed on unit 2 in stage 1.
The processing sequence on the unit in stage 2 is CDBA.
A A
B B
C C
D D
Stage 1 Tahap 1
C C
D D
Stage 2 Tahap 2
B B
A A
Unit 1
Unit 2
Unit 1
Figure 9:
Solution Representation - Example
Once we have a feasible allocation and sequencing of the orders in various units, the start
and end times of the products in the stages can be evaluated with expressions similar to (45) and
(46). The makespan may be calculated as in (47) and the area compensation applied to the fuzzy
makespan to represent the objective value. The makespan of a given solution may be computed in
Ô‰ |®p
²
p p
time. waktu.
6.4.2 Neighborhood of a Solution
We consider insertion moves, where we select a product
@ @
– -
on a unit in stage
Ï Aku
– -
and insert it in
another position on the same unit or any other unit on the same stage. Clearly, then the neigh-
borhood defined by the insertion move is of size
Ô— c®
™ ™
p p
²
, since
Ï Aku
– -
can be chosen in M ways,
@ @
– -
can be chosen in
® ®
ways, and the position of insertion can be chosen in
Ô‰ |®D
ways. cara. Since Sejak
the computation of the makespan of a given solution takes
Ô‰ |®ûp
²
p p
time, the evaluation of
the complete insertion neighborhood takes
Ô‰ |®T×íp
²
™ ™
p p
time. waktu. Figures 10 and 11 display two
possible insertion moves from the solution in Figure 9 for the 4-product 2-stage example discussed
earlier. sebelumnya. Figure 10 shows product A being removed from Unit 1 on Stage 1 and inserted ahead of
products C and D on Unit 2 in the same stage, while Figure 11 shows the insertion of product D at
the end of the processing sequence on Unit 1 in Stage 2.
The computational complexity of evaluating the complete insertion neighborhood was con-
firmed by tests on a number of problems, with the result that evaluating the entire insertion neigh-
borhood was deemed too time-consuming. As a result, we used a reduced neighborhood based on
the insertion move. The gains in computational efficiency for the reduced neighborhood in compar-
ison with the entire neighborhood are quite significant. For instance, the CPU time for one iteration
of a 25-product, 5-stage, 15-machine problem with a full evaluation of the insertion neighborhood
22 22
Page 23 Page 23
Stage 1 Tahap 1
A A
B B
C C
D D
Unit 1
Unit 2
Figure 10:
Insertion Move - Different Units
C C
D D
Stage 2 Tahap 2
B B
A A
Unit 1
Figure 11:
Insertion Move - Same Unit
was 4.9 secs, while for the reduced neighborhood implementation, the CPU time for one iteration
was 0.50 secs. Please refer to Appendix A for a discussion of the reduced neighborhood.
6.4.3 Computational Results
To see how well the RTS algorithm performs on the multistage parallel flowshop problems, the 10-
product, 5-stage, 15-units problem solved in Section 6.3 was solved with a limit of 50000 iterations
(about 2800 CPU secs).
Recall that for model (M1), even after 50,000 CPU secs of solution time, there was an opti-
mality gap of over 50% (ie, the best solution possible was 113.38 and the best solution obtained
was 250.086), while for (M2), the best solution obtained had an objective value of 280.932. It Ini
is likely that the large number of binary variables and the Big-M constraints in the MILP model
result in there being such poor LP relaxations: thus, the best solution possible may have a signifi-
cantly larger objective value than that indicated when the branch and bound algorithm terminated.
Indeed, fixing the ordering of a subset of products on even one unit leads to much better bounds
on the objective.
Using the RTS algorithm, the best solution (obtained in 2800 CPU secs) had a makespan of
182.878. Note that the RTS algorithm obtained a much better solution in a fraction of the time
required by the mathematical programming approach. The key observation that we can derive
from this example is that local-search techniques such as the RTS algorithm are well suited for
solving the parallel flowshop problems since they are able to provide good quality solutions with
reduced computational requirements. However, MILP models are not completely useless for these
23 23
Page 24 Page 24
problems, since they can be used to derive bounding information that indicate the quality of the
solutions obtained using the local search techniques.
300 300
400 400
500 500
600 600
700 700
800 800
900 900
1000 1000
0 0
5000 5000
10000 10000
15000 15000
20000 20000
25000 25000
Objective Function
Iteration
Solutions Found Over RTS Progress
Current Solution
Best Solution so far
Figure 12:
RTS Progress
Another advantage with techniques such as RTS is that they usually scale quite well with
problem size. Often, the iteration complexity grows only polynomially with problem size. Figure Gambar
12 displays the progress of an RTS implementation for a 25-product, 5-stage, 15-machine problem.
The RTS algorithm used the reduced neighborhood of the insertion move, as described earlier; the
CPU time for 25,000 iteration was 11,600 secs, thus averaging under 0.5 secs per iteration. Clearly
then, our RTS implementation shows promise for tackling even bigger problems. The starting so-
lution which was obtained by a random balanced allocation of the jobs to the various units in the
stages had an objective value of 776.197 units. The best solution (shown below) obtained had a
value of 326.236 units and was obtained at iteration 2408. Thus, on unit 1 in stage 1, product
… ý
is adalah
processed first, and is followed by products
… ...
× ×
, ,
… ...
–|þ
etc. It is instructive to note from Figure 12 that
this solution was obtained soon after one of the random escapes was performed, thereby validating
the utility of the escape sequences in the RTS algorithm.
Best Solution obtained from the RTS algorithm
24 24
Page 25 Page 25
Stage 1 Tahap 1
Unit 1:
… ý–ÿ
… ...
× ×
ÿ ÿ
… ...
–|þ
ÿ ÿ
… ...
– -
× ×
ÿ ÿ
… Ñ–ÿ
… ...
– -
ýíÿ
… ...
–¡
ÿ ÿ
… ...
— -
ÿ ÿ
… ™ÓÒGÿ
… ...
– -
Ò O
Unit 2:
…£¢ ÿ
… ...
–z–
ÿ ÿ
… ...
– -
ÿ ÿ
… ...
– -
™Gÿ
… ™–ÿ
… ™
– -
Unit 3:
… ...

ÿ ÿ
… ...
– -
¢–ÿ
… ...
– -
— -
ÿ ÿ
…C™z™Gÿ
… ...
þ Þ
ÿ ÿ
…C™
× ×
ÿ ÿ
…CÒGÿ
…C™
— -
ÿ ÿ
… ...
– -
Ñ Ñ
Stage 2 Tahap 2
Unit 1:
…£¢ ÿ
… ...
× ×
ÿ ÿ
… ...
–z–
ÿ ÿ
… ...
– -
— -
ÿ ÿ
… ...
þ Þ
ÿ ÿ
… ...
– -
× ×
ÿ ÿ
… ™z™–ÿ
…C™
× ×
ÿ ÿ
… ™
— -
ÿ ÿ
…$Òíÿ
… ...
– -
Ñ–ÿ
… ...
– -
Ò O
Unit 2:
… ý–ÿ
… ...

ÿ ÿ
… ...
– -
¢ ÿ
… ...
–|þ
ÿ ÿ
… ...
– -
™–ÿ
… ...
– -
ÿ ÿ
… Ñ–ÿ
… ...
– -
ýíÿ
… ...
— -
ÿ ÿ
… ™ÓÒGÿ
… ™
– -
ÿ ÿ
… ...
–¡
ÿ ÿ
… ™
Stage 3 Tahap 3
Unit 1:
…£¢ ÿ
… ...
× ×
ÿ ÿ
… ...
–z–
ÿ ÿ
… ...
– -
ÿ ÿ
… ...
– -
— -
ÿ ÿ
… ...
– -
ýíÿ
… ...
— -
ÿ ÿ
…CÒGÿ
… ...
– -
Ò O
Unit 2:
… ...
–|þ
ÿ ÿ
… ...
þ Þ
ÿ ÿ
… Ñ–ÿ
…C™ÓÒíÿ
… ...
–¡
Unit 3:
… ý–ÿ
… ...

ÿ ÿ
… ...
– -
¢ ÿ
… ...
– -
™Gÿ
… ...
– -
× ×
ÿ ÿ
…C™
× ×
ÿ ÿ
… ™
— -
ÿ ÿ
…C™z™Gÿ
… ...
– -
ÑGÿ
…C™
– -
ÿ ÿ
… ™
Stage 4
Unit 1:
… ...
– -
ÿ ÿ
…CÑGÿ
… ™
— -
ÿ ÿ
… ...
— -
ÿ ÿ
…CÒGÿ
… ...
– -
Ò O
Unit 2:
… ...
– -
¢–ÿ
…C™ÓÒ
Unit 3:
… ...
× ×
ÿ ÿ
… ...
–|þ
ÿ ÿ
… ...
– -
™Gÿ
… ...
– -
× ×
ÿ ÿ
… ™
× ×
ÿ ÿ
…C™
– -
ÿ ÿ
… ™
Unit 4:
… ý–ÿ
… ...

ÿ ÿ
…£¢ ÿ
… ...
þ Þ
ÿ ÿ
… ...
–z–
ÿ ÿ
… ...
– -
— -
ÿ ÿ
… ...
– -
ý–ÿ
… ™z™íÿ
… ...
– -
Ñ–ÿ
… ...
–¡
Stage 5
Unit 1:
…£¢ ÿ
… ...
– -
ÿ ÿ
… ...
– -
™–ÿ
…C™
— -
ÿ ÿ
… ™z™–ÿ
…C™
– -
ÿ ÿ
… ™
Unit 2:
… ...
–z–
ÿ ÿ
… ...
– -
× ×
ÿ ÿ
… ...
– -
— -
ÿ ÿ
… ...
– -
ýGÿ
… ...
— -
ÿ ÿ
… ...
–¡
ÿ ÿ
… ...
– -
Ñ Ñ
Unit 3:
… ý–ÿ
… ...

ÿ ÿ
… ...
– -
¢ ÿ
… ...
–|þ
ÿ ÿ
… ...
× ×
ÿ ÿ
… ...
þ Þ
ÿ ÿ
… Ñ–ÿ
… ™
× ×
ÿ ÿ
… ™ÓÒGÿ
…CÒGÿ
… ...
– -
Ò O
Figure 13 displays the variation of the size of the tabu list over the course of the RTS. The The
tabu list size typically oscillates between 5 and 60 as the search varies between intensification and
diversification.
Although we have only illustrated the RTS implentation for the makespan minimization prob-
lem, extending it to lateness minimization is a straightforward task, requiring only minor changes
in the computations required at each iteration.
7 The New Product Development Problem
In the NPD process, the potential product must pass a series of tests that assess its safety, efficacy
and environmental impact. The costs, durations and probabilities of success of various testing tasks
are often not known with certainty when the schedule has to be constructed. The optimization of
the testing schedule can be conceived as introducing precedence constraints between the tests, in
addition to the given set of technological precedence constraints. There is an inherent cost-income
trade-off associated with the scheduling of the NPD process. This trade-off is between the greater
25 25
Page 26 Page 26
0 0
10 10
20 20
30 30
40 40
50 50
60 60
70 70
0 0
5000 5000
10000 10000
15000 15000
20000 20000
25000 25000
Tabu List Length
¤ ¤
Iteration
Length of Tabu List Over RTS Progress
Figure 13:
Tabu List Size
income resulting from a shorter schedule (where many of the tests are performed in parallel) and
the lower expected value of the total cost from a longer sequential schedule.
We consider the problem of optimizing the schedule of testing tasks while taking into account
the uncertainties in the durations and success of the tasks, a problem which has not been reported
before. sebelumnya. In previous work, Schmidt and Grossmann (1996) developed an MILP model assuming
known durations. These authors also considered the evaluation of the distribution of the completion
time for a fixed schedule, given uncertainties in durations (Schmidt and Grossmann, 2000). The The
extension of their method to optimization is non-trivial, given the complexity of their procedure
for evaluating the completion time with uncertain durations.
To effectively address the problem, we use a two-level description of the uncertainties inher-
ent in the NPD process. We retain the probability description of the success of the tests, ie, for
every test there is a given probability that the product will pass the test. However, instead of as-
suming the discrete/continuous probability distributions that Schmidt and Grossmann (2000) used
to model the uncertainty in test durations, we draw upon concepts of fuzzy set theory to represent
the imprecision and uncertainty of information in the time parameters of the various activities.
Thus, the processing times are represented by intervals or fuzzy numbers instead of deterministic
26 26
Page 27 Page 27
values or probability distributions. For a given schedule, the completion time is computed through
interval arithmetic by the extension principle. This computation can be performed efficiently, in
contrast to the case when we use a probabilistic description of the testing durations. The expected Yang diharapkan
value of the cost of a schedule is evaluated using the probabilistic framework, and associated with
this successful schedule is a fuzzy description of the project's completion time. This completion
time is then translated into an equivalent income description, thereby completing the evaluation of
a given schedule in terms of the expected cost and the completion time (income).
The optimization of the schedule considers as the objective the difference of the expected
value of the fuzzy income and the expected cost. Formulation (PDP) represents a MILP model for
selecting the optimal schedule under task uncertainty. It is assumed that the reader is familiar with
the deterministic version of the problem (Schmidt and Grossmann, 1996).
(PDP) Max
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(58) (58)
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(59)
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(60) (60)
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In formulation (PDP), the precedence relationship between two tasks
and is modeled with
the help of the binary variable
wdÂ
¹
. . Thus Demikian
wEÂ
¹
e e
if task
precedes task , else
wdÂ
¹
H H
. . In Dalam
(PDP), (54) represents the objective function, which is calculated as the difference of the maximum
possible income,
²
± ±
, and the cost of the schedule and the total decrease in income due to delay
27 27
Page 28 Page 28
in product release. Since the durations of the various testing tasks are given by fuzzy numbers,
the schedule completion time and the decrease in income are also fuzzy numbers. The mean value
of the total decrease in income as calculated by the area compensation integral is included as part
of the objective function. The The
¤œ€
s denote the Simpson coefficients used in the approximation of
the integral;
¯öÄ
denote the points in time when the income decrease rate
±P†'Ä
changes and
… ¹
indicates the probability of success of task . Constraints (55) forces the completion time,
·y“
, (at
every setiap
Q Q
-level) of the schedule to be greater than the completion time of each task (at every
Q Q
-level),
while (56) aid in the calculation of the excess time over
¯¨Ä
. . The timing relationship between two
tasks
and is modeled using the Big-M formulation in (57), which specifies that either task
starts after
is complete, or task starts before
, or the relationship between the two tasks is
unspecified. Constraints (58) fix the technological precedence relationships (given by set
… ...
s s
€ €
) )
between certain tasks
and , while (59) and (60) approximate the exponential in calculating the
cost term. Finally, (61)-(63) are logic cuts and valid inequalities that eliminate directed cycles
of length 2 or 3, that strengthen the relaxation of the MILP. We now present two examples that
illustrate the performance of model (PDP).
7.1 The 9-Task Example
The data for this example are taken from Schmidt and Grossmann (1996) and are modified to re-
flect the uncertainty in testing durations. In particular, the testing success probabilities and most
likely values of the durations are the same as those in the above paper. However, the durations are
assumed to be given by asymmetric TFNs, with a left spread of 5% and a right spread of 20%.
Figure 14 displays the technological precedences for this problem, while Figure 15 displays the
optimal schedule when the computations are performed with 20 intervals (21 points). Computa-
tional results from two discretizations are shown in Table 5. Both models provide the same optimal
solution, although the 21-pt discretization model provides a better estimate of the mean value of
the profit. The number of equations and continuous variables in the 21-pt model is 160% and 85%
more than the 7-pt model and the solution time is an order of magnitude larger. However, the Namun,
improvement in the estimation of the income is less than 1%, which does not really warrant the
order of magnitude increase in the computational time.
7.2 The 65-Task Example
The task data for this example are taken from Schmidt (1998) and are modified to reflect the
uncertainty in testing durations. The durations are assumed to be given by asymmetric TFNs,
with a left spread of 5% and a right-spread of 20% and with the central value at the data-point in
28 28
Page 29 Page 29
H H
G G
E E
Figure 14:
Technological Prece-
dences
D D
I Aku
G G
F F
C C
E E
B B
H H
A A
Figure 15:
Optimal Schedule
Table 5: Computational Results for the 9-Task Example
Model Characteristics
7-pt Model 21-pt Model
Equations
1402
3754
Variables Variabel
492 492
856 856
Binaries Binari
70 70
70 70
CPU secs
601 601
7611
Schedule Duration (as TFN)
(19, 20, 24)
(19, 20, 24)
Schedule Cost ($1000s)
1033.6445
1033.6445
Income Loss (Mean) ($1000s) 1156.9444
1120
Profit ($1000s)
3809.4110
3846.355
Schmidt (1998). The maximum income is $25 million and the income decrease is over a period
of 72 months. Figure 16 displays the technological precedences between the various tasks, while
Figure 17 shows the optimal schedule obtained by solving the MILP with a 7-pt discretization.
The other models with finer discretization provided the same optimal solution, although with better
estimates of the objective function. These models solve comparatively faster than the smaller 9-
task example because there are a number of technological precedence constraints (see Figure 16),
which leads to more structure in the schedule. The number of equations and continuous variables
in the 21-pt model is 80% and 70% more than the 7-pt model and the solution time is almost an
order of magnitude more. The only benefit of using the 21-pt model is a more accurate estimation
of the duration, income (2% higher than the 7-pt model). Similarly, the 51-pt discretization model
provides less than a 1% improvement in the estimation of the income loss when compared with the
21-pt model, although the solution time has increased by a factor of 3.
29 29
Page 30 Page 30
29 29
32 32
28 28
31 31
30 30
27 27
59 59
62 62
58 58
26 26
57 57
61 61
25 25
24 24
56 56
65 65
23 23
55 55
22 22
54 54
53 53
63 63
21 21
20 20
52 52
60 60
51 51
50 50
9 9
13 13
8 8
12 12
19 19
18 18
7 7
49 49
17 17
6 6
16 16
5 5
48 48
47 47
4 4
15 15
14 14
3 3
46 46
2 2
45 45
1 1
11 11
10 10
44 44
43 43
42 42
41 41
40 40
39 39
38 38
64 64
37 37
36 36
35 35
34 34
33 33
Figure 16:
Technological Prece-
dences
29 29
32 32
28 28
31 31
27 27
59 59
58 58
56 56
55 55
38 38
35 35
62 62
26 26
30 30
57 57
61 61
25 25
24 24
65 65
23 23
54 54
53 53
43 43
22 22
63 63
21 21
20 20
52 52
60 60
51 51
50 50
47 47
40 40
9 9
13 13
19 19
8 8
12 12
7 7
18 18
17 17
6 6
49 49
16 16
5 5
48 48
15 15
4 4
11 11
10 10
14 14
3 3
46 46
2 2
45 45
1 1
44 44
42 42
41 41
39 39
64 64
37 37
36 36
34 34
33 33
Figure 17:
Optimal Schedule
Table 6: Computational Results for the 65-Task Example
Model Characteristics
7-pt Model
21-pt Model
51-pt Model
Equations
148780
267192
520932
Variables Variabel
6834
8794
12994
Binaries Binari
4061 4061
4061 4061
4061 4061
CPU secs
324 324
2786
8909
Schedule Durn (TFN)
(58.11,61.17,73.4) (58.11,61.17,73.4) (58.11,61.17,73.4)
Schedule Cost ($1000s)
3147.2161
3147.2161
3147.2161
Income Loss (Mean) ($1000s)
8487.9514
8194.8833
8119 8119
Profit ($1000s)
13364.8326
13657.9006
13732.86
30 30
Page 31 Page 31
8 Conclusions
In this paper we have applied a non-probabilistic approach to the treatment of processing time
uncertainty in two problems - the scheduling of (i) flowshop plants and (ii) the new product devel-
opment process. The benefits of using the non-probabilistic approach are as follows:
1. 1. Concepts such as fuzzy sets, interval arithmetic allow us to model uncertainty in cases
where historical (probabilistic) data are not readily available. These models are fairly flexible in
their description of uncertainty - for example, different types of functions can be used to model
the uncertainty in the parameters. Here we have used only Triangular Fuzzy Numbers (TFNs) to
capture the uncertainty in the task durations. An advantage of this representation is the ease of
interpretation of the results, for instance, in terms of the most likely, optimistic and pessimistic
estimates of the makespan, although an interpretation of the objective function value is not as
straight forward as in the probabilistic case (for eg, in terms of the expected value).
2. 2. The most significant gain is the computation time required to solve the optimization mod-
els under uncertainty. These models neither suffer from the combinatorial explosion of scenarios
that discrete probabilistic uncertainty representations exhibit, nor do they require complicated in-
tegration schemes which continuous probabilistic models require. The MILP models developed in
this work could be solved with reasonable computational effort for problems where the solution
was more structured. These models did not perform as well when we attempted to solve general
problems to optimality.
3. 3. Another significant advantage of the fuzzy-set approach (over the probabilistic approach)
is that heuristic search algorithms for combinatorial optimization such as tabu search can be easily
applied to obtain good quality solutions in reasonable computing time. This is primarily due to the
ease in computing the objective function of a solution.
4. 4. The examples considered show that very good estimates of the uncertain makespan and
income can be obtained by using fairly coarse discretizations and that these models can be solved
with little computational effort. In these examples the improvement in the estimation of the com-
pletion time by using a denser discretization was not significant enough to warrant the order of
magnitude increase in computation time required.
Appendix A - Reduced Neighborhood
The idea of using a reduced neighborhood to improve the computational characteristics of local
search algorithms is not new. For instance, van Laarhoven et al. (1992) used the concept of critical
arcs in jobshop scheduling problems for eliminating moves that will definitely not improve the
31 31
Page 32 Page 32
solution. solusi. This idea was then extended to the fuzzy jobshop scheduling problem by Fortemps
(1997). (1997). A recent paper by Nowicki and Smutnicki (1998) also employs a specific neighborhood
definition based on critical paths for flow shops with parallel machines in a deterministic setting.
With this reduced neighborhood and further advanced implementations of tabu list querying and
searching, the authors were able to dramatically improve the speed of their tabu search algorithm.
In our implementation, we build upon the ideas in both Fortemps (1997) and Nowicki and
Smutnicki (1998) to define our reduced neighborhood. Thus, for a given solution, we identify the
critical path(s) and consider only the insertion of jobs that lie on the critical path(s). A move that
seeks to insert a job that does not lie on any of the critical paths is considered useless, in that it
will not improve the makespan. The main difference between the deterministic version and our
case is that there may be many more critical jobs, since different paths may be critical at different
Q Q
-levels. The identification of the critical path(s) for a solution is a straightforward operation and
does not contribute significantly to the computational complexity of an iteration.
However, we have to account for the memory required for storing the additional information
about the critical jobs with every solution. If we store information about all critical paths at all
Q Q
-levels, the memory required becomes excessive very quickly. Therefore, in our implementation
we have stored the predecessor information to be used for computing the critical path from only the
Q Q
e e
level, which corresponds to the most possible realization of processing times. Checking if
a move is useless is done in constant time using a hashtable containing the critical jobs associated
with the solution. Fortunately, with these algorithmic improvements, the size of the reduced neigh-
borhood is only
Ô‰ c®žp "
, where , Dimana
is the number of jobs on the critical path. Now, Sekarang,
is often
much smaller than
®¼p
²
, thus when compared with the
Ô‰ |®
™ ™
p p
²
size of the complete insertion
neighborhood we have a considerable improvement. Also, the solutions obtained by performing
the useless moves do not need to be evaluated in the reduced neighborhood implementation unlike
the complete implementation, thereby leading to a further reduction in the computational effort.
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